1 CHAPTER-I INTRODUCTION 1.1 Groundwater Groundwater is water located under the ground surface in soil pore spaces and in the fractures of rock formations. Groundwater recharge can be made due to natural rainfall however natural discharge often occurs at springs and seeps, and can form oases or wetlands. Groundwater is often withdrawn for agricultural, municipal and industrial use by means of constructions and operations of extraction wells. The study of distribution and movement of groundwater in hydrogeology is called groundwater hydrology. The water cycle, also known as the hydrologic cycle describes the continuous movement of water on, above and below the surface of the Earth. The hydrologic cycle is used to model the storage and movement of water between the biosphere, atmosphere, lithosphere and hydrosphere. Water is stored in the reservoirs such as atmosphere, oceans, lakes, rivers, glaciers, soils, snowfields, and groundwater. Fig. 1.1 The hydrological (water) cycle
Transcript
1
CHAPTER-I
INTRODUCTION
1.1 Groundwater
Groundwater is water located under the ground surface in soil pore spaces and in
the fractures of rock formations. Groundwater recharge can be made due to natural rainfall
however natural discharge often occurs at springs and seeps, and can form oases or
wetlands. Groundwater is often withdrawn for agricultural, municipal and industrial use by
means of constructions and operations of extraction wells. The study of distribution and
movement of groundwater in hydrogeology is called groundwater hydrology. The water
cycle, also known as the hydrologic cycle describes the continuous movement of water on,
above and below the surface of the Earth. The hydrologic cycle is used to model the storage
and movement of water between the biosphere, atmosphere, lithosphere and hydrosphere.
Water is stored in the reservoirs such as atmosphere, oceans, lakes, rivers, glaciers, soils,
snowfields, and groundwater.
Fig. 1.1 The hydrological (water) cycle
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It moves from one reservoir to another by processes like: evaporation, condensation,
precipitation, runoff, infiltration, transpiration, and groundwater flow as shown in Fig.1.1.
The water moves from one reservoir to another, such as from river to ocean, or from the
ocean to the atmosphere, by the physical processes of evaporation, condensation,
precipitation, infiltration, runoff, and subsurface flow. The water goes through different
phases like liquid, solid, and gas during these process and the key elements are discussed
below:
Evaporation: Evaporation is the process of a liquid becoming vaporized. In other words, a
change in phase in the atmosphere occurs when substances change from a liquid to a
gaseous, or vapor, form. Evaporation in the atmosphere is a crucial step in the water cycle.
Condensation: The physical process by which a vapor becomes a liquid or solid, the
opposite to evaporation is known as condensation. In meteorological usage, this term may
be applied for the transformation from vapor to liquid. Condensation commonly occurs
when a vapor is cooled and/or compressed to its saturation limit when the molecular
density in the gas phase reaches its maximal threshold.
Transpiration: It is the process by which moisture is carried through plants from roots to
small pores on the underside of leaves, where it changes to vapor and then released to the
atmosphere.
Precipitation: It is the process of atmospheric discharge of water in the solid or liquid state
on the earth surface. The various forms of precipitation are: rain, drizzle, snow, snow
grains, snow pellets, diamond dust, hail, and ice pellets etc.
Infiltration: It is the process by which water on the ground surface enters into the soil.
Infiltration rate in soil science is nothing but the rate at which soil is able to absorb rainfall
or irrigation. The rate of infiltration is affected by soil characteristics including ease of
entry, storage capacity, and transmission rate through the soil. The soil texture and
structure, vegetation patterns, water content, soil temperature, and rainfall intensity all
these play a significant role in controlling the infiltration rate and capacity. For example,
coarse-grained sandy soils have large spaces between each grain and allow water to
infiltrate quickly.
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1.2 Runoff:
The portion of rainfall, which makes its way towards streams, rivers etc. after
satisfying the initial losses etc. is known as runoff. The runoff may be classified as follows:
Fig. 1.2 Different flow types (Musy, 2001)
Surface Runoff: It is that portion of rainfall which enters the stream immediately after the
rainfall. It occurs when all losses are satisfied and if rain is still continued, with the rate
greater than infiltration rate then at this stage the excess water makes a head over the
ground surface (surface detention). This tends to move from one place to another is known
as overland flow. As soon as the overland flow joins to the streams, channels or oceans,
termed as surface runoff.
Sub-surface Runoff: The part of rainfall, which first leaches into the soil and moves
laterally without joining the water-table to the streams, rivers or oceans is known as sub-
surface runoff.
Base Flow: It is delays flow, defined as that part of rainfall which after talling on the
ground surface infiltrated into the soil and meets the water table and flow to the streams
oceans etc. The movement of water in this type of runoff is very slow and therefore it is
also known as delayed runoff. It takes a long time to join the rivers or oceans. Sometimes
base flow is also known as groundwater flow.
Thus, Total Runoff = Surface runoff + Base flow (Including sub - surface runoff)
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1.3 Vertical Distribution of Groundwater:
Water in the subsurface may be divided into two major zones: i) water stored in the
unsaturated zone also known as vadose zone or zone of aeration and ii) water stored in the
saturated zone. Soil pore spaces in the vadose zone, lying immediately below the surface.
Here the small pore spaces between soil particles are filled with a mixture of water and air
resulting in an area which is less than saturated zone. This zone may be divided with
respect to occurrence and circulation of water into the uppermost zone of soil water, the
intermediate zone and the capillary fringe, immediately above the water table. Water in this
zone is called capillary water. This water moves upward from the water table by capillary
action. Capillary water moves slowly in any direction. Water cannot be withdrawn from
this zone for residential or commercial water supply purpose because the capillary forces
hold it too tightly. The roots of trees, plants and crops, however, can tap into this water.
The capillary fringe moves upwards and downwards together with the water table due to
seasonal pattern. Fig 1.3a and 1.3b shows the distribution of water in the subsurface
regions.
Groundwater is water below the water table, falling entirely all rock interstices
(void spaces) in the saturated zone. The water located in this zone can be withdrawn for
various uses. The variation in the flow of groundwater depends on the type of rocks or
other permeable material, the size of the pore spaces in the soil or rock, connectivity of
pore spaces, and the configuration of the underground strata.
Water Table: The upper surface of the zone of saturation is known as water table. At the
water table, the water in the pores of the aquifer is at atmospheric pressure. The hydraulic
pressure at any level within a water table aquifer is equal to the depth from the water table
point and is referred to as the hydraulic head. When a well is dug in a water table aquifer,
the static water level in the well stands at the same elevation as the water table. The
groundwater table, sometimes called the free or phreatic surface, is not a stationary surface.
This water table moves up and down due to various reason. It may rises when more water is
added to the saturated zone by vertical percolation, and drops down during drought periods
when the stored water flows out towards springs, streams, well and other points of
groundwater discharge.
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1.3a Subsurface distribution of water
Fig. 1.3b Subsurface distribution of water
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1.4 Groundwater Contamination:
Groundwater contamination occurs when contaminants are discharged directly or
indirectly into water bodies without adequate treatment to remove harmful compounds. The
widespread use of chemical products, coupled with the disposal of large volumes of waste
materials, poses the potential for widely distributed groundwater contamination. Hazardous
chemicals such as pesticides, herbicides, and solvents, are used everywhere in everyday
life. These and a host of other chemicals are in widespread use in urban, industrial, and
agricultural areas. The largest potential source of groundwater contamination is the disposal
of solid and liquid wastes. Waste disposal is not only the source of groundwater
contamination but some additional sources like septic tank systems, agriculture, accidental
leaks and spills, mining, artificial recharge, underground injection, and saltwater
encroachment etc. are causes for groundwater contamination.
1.5 Sources of Groundwater Contamination:
The contaminants can be introduced in the groundwater by means of natural
occurring activities, such as natural leaching of the soil and mixing with other groundwater
sources having different chemistry. These all are also introduced by planned human
activities, such as waste disposal, mining, and agricultural operations. Almost every major
industrial and agricultural site has in the past disposed of its wastes on site, often in an
unnoticeable location. As we all know every municipality has had to dispose of its waste at
selected locations within its proximity. Accidental spills of toxic chemicals have also
occurred, often without particular attention to or concern for the consequences. Some
practices of cleaning a toxic spill involve flushing it with water until it disappears into the
ground. Past waste-disposal practices and dealing with spills have not always considered
the potential for groundwater contamination. Factories and underground storage tanks are
also a source of groundwater contaminantion. If a tank with water soluble liquid leaks the
liquid travels down to the water table and then it dissolves in the groundwater. These
pollutants flow as a plume along with the groundwater which can pollute wells and the
plume path shown below in Fig. 1.4.
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Fig. 1.4 Sources of pollution
A problem of growing concern is the cumulative impact of contamination of aquifer from
non-point sources such as those created by intensive use of fertilizers, herbicides, and
pesticides. In addition, small point sources-such as numerous domestic septic tanks or small
accidental spills from both agricultural and industrial source threaten the quality of
aquifers. If the liquid that leaks is less dense than water, it floats on top of the groundwater
table. Some of the liquid evaporates and travels upwards to the surface in the form of vapor
fumes. Some of the liquid dissolves and travels as a plume in the groundwater. If the
chemical is not very soluble in water then the major part of the liquid floats on the
groundwater and flow along with the groundwater. With respect to mixing tendency the
sources of contamination may be classified as follows:
Point Sources: Point source contamination represents those activities where wastewater is
routed directly into receiving water bodies. Point source contaminants in groundwater are
usually found in a plume that has the highest concentration of the contaminants near by the
source and diminishes concentration farther away from the source. The example of point
source contaminantion can be enlisted as follows:
Direct discharges from factories: raw materials and wastes may include pollutants
such as solvents, petroleum products (e.g., oil and gasoline), or heavy metals etc.
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Agriculture may also include point source contamination such as animal feeding
operations, animal waste treatment lagoons, or storage, handling, mixing, and
cleaning areas for pesticides, fertilizers and petroleum etc.
Municipal point sources may include wastewater treatment plants, landfills,
garages, motor pools and fleet maintenance facilities etc..
Other sources include mine discharge water and mine spoil run-off etc.
Non-Point Sources: Non-point source contamination represents those activities where
wastewater is routed indirectly into receiving water bodies due to environmental changes.
The discharge from nonpoint sources are usually intermittent, associated with a rainfall or
snowmelt event, and occur less frequently and for shorter periods of time than the
discharges from point sources. Nonpoint sources of contamination are often difficult to
identify, isolate and control. The example of non-point source contamination can be
enlisted as follows:
i) Automobile emissions, road dirt and grit, and runoff from parking lots etc.
ii) Runoff and leachate from agricultural fields, barnyards, feedlots, lawns, home gardens
and failing on-site wastewater treatment systems etc. and
iii) Runoff and leachate from construction, mining and logging operations etc.
1.6 Aquifer and its Types:
Aquifer: The word aquifer comes from two Latin words: aqua (water) and affero (to bring
or to give). An Aquifer is geological formation, part of a formation or group of formations
that contain sufficient saturated permeable material to yield significant quantities of water
to well and springs. Water within the zone of saturation is at a pressure greater than
atmospheric pressure. Sand and gravel deposits, sandstone, limestone, and fractured,
crystalline rocks are examples of geological units that form aquifers.
Aquitard: Aquitard is closely related to aquifer, is also derived from the two Latin words:
aqua (water) and tardus (slow) or tardo (to slow down, hinder, delay). This means that
aquitard does store water and is capable of transmitting it, but at a much slower rate than an
aquifer so that it cannot provide significant quantities of potable groundwater to well and
springs;e.g., sandy clay.
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Aquiclude: Aquiclude is another related term, is also derived from two Latin words: aqua
(water) and claudo (to confine, close, make inaccessible). Aquiclude is equivalent to an
aquitard of very low permeability, which for all practical purposes; act as an impermeable
barrier to groundwater and contaminant flow; clay is an example.
Aquifuse: It is a relatively impermeable formation with no interconnected pores and hence
neither containing nor transmitting water. It has very low porosity and very low
permeability. For example, hard rock formations such basalts and granites which are free
from fractures, faults or weathering.
Aquifer may be broadly classified as follows:
Confined Aquifer: A confined aquifer is located between layers of impermeable materials
that restrict the flow of water into out of the aquifer. The pressure in this type of aquifer is
high due to the confining layers that enable the water level in wells to rise above the typical
water level of an aquifer. The pressure condition in a confined aquifer is characterized by a
piezometric surface, which is the surface obtained by connecting equilibrium water levels
in tubes or piezometers penetrating the confined layer.
Unconfined Aquifer: An unconfined aquifer is a layer of water-bearing material without a
confining layer at the top of the groundwater, called the groundwater table, where the
pressure is equal to atmospheric pressure. The groundwater table, sometimes called the free
or phreatic surface, is free to rise or fall. The groundwater table height corresponds to the
equilibrium water level in a well penetrating the aquifer. Above the water table is the
vadose zone, where water pressures are less than atmospheric pressure. The soil in the
vadose zone is partially saturated, and the air is usually continuous down to the unconfined
aquifer.
Perched Aquifer: A perched aquifer refers to groundwater that is separated from the
underlying main body of groundwater, or aquifer, by unsaturated rock or confine rock.
The different types of aquifers discussed above are shown below in Fig. 1.5.
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Fig. 1.5 Types of aquifers
1.7 Characteristics of Aquifer:
Porosity: The void space in between the crystals or fragments that make up a rock
represent porosity that can hold water. Porosity of an aquifer is the percentage of void
spaces occupied by water or air in the total volume of rock which includes both solids and
voids
100%vVV
(1.1)
where vV = The volume of all rock voids and
V = The total volume of rock.
Porosity can also be expressed as
1 100%m d d
m m
(1.2)
where m = Average density of minerals particles
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d = Density of dry sample.
Porosity is the most important property of rocks that enable storage and movement of water
in the subsurface. It directly influences the permeability and the hydraulic conductivity of
rocks and therefore the velocity of groundwater and other fluids that may present. In
general, rock permeability and groundwater velocity depend on the shape, amount,
distribution and interconnectivity of voids. On the other hand, voids depend on the
depositional mechanisms of unconsolidated and consolidated sedimentary rocks, and on
various other geologic processes that affect all rocks during and after their formations.
Primary porosity is the porosity formed during the formation of rock itself, such as voids
between the grains of sand, voids between minerals in hard (consolidated) rocks, or
bedding planes of sedimentary rocks. Secondary porosity is created after the rock
formation mainly due to tectonic forces (faulting and folding), which creates micro- and
macro- fissures, fractures, faults and fault zones in the solid rocks.
Factor Affecting the Magnitude of Porosity:
In sediments or sedimentary rocks the porosity depends on grain size, the shapes of the
grains, and the degree of sorting, and the degree of cementation.
(i)
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(ii) (iii)
Fig. 1.6 Factor affecting the porosity of a sedimentary rock
i) Well-rounded coarse-grained sediments usually have higher porosity than fine-grained
sediments, because the grains do not fit together well.
ii) Poorly sorted sediments usually have lower porosity because the fine-grained fragments
tend to fill in the open space.
iii) Since cements tend to fill in the pore space, highly cemented sedimentary rocks have
lower porosity.
Fig. 1.7 Types of rock intensities and the relation of rock texture to porosity
Types of porosity with relation to rock texture
a) Well-sorted sedimentary deposit having high porosity.
b) Poorly sorted sedimentary deposit having low porosity.
c) Well-sorted sedimentary deposit consisting of pebbles that are themselves porous, so that
the deposit as a whole has a very high porosity.
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d) Well-sorted sedimentary deposit whose porosity has been diminished by the deposition
of mineral matter in the interstices.
e) Rock rendered porous by solution.
f) Rock rendered porous by fracturing.
Permeability: Permeability is a measure of the degree to which the pore spaces are
interconnected, and the size of the interconnections. Low porosity usually results in low
permeability, but high porosity does not necessarily imply high permeability. It is possible
to have a highly porous rock with little or no interconnections between pores. A good
example of a rock with high porosity and low permeability is a vesicular volcanic rock,
where the bubbles that once contains gas give the rock a high porosity, but since these
pores are not connected to one another the rock has low permeability. It depends only on
the physical properties of the porous medium, grain size, grain shape and arrangement, pore
interconnection etc.
Fig. 1.8 Characterization of permeability
A thin layer of water is always attract to mineral grains due to the unsatisfied ionic charge
on the surface. This is called the force of molecular attraction shown in Fig. 1.8. If the size
of interconnections is not as large as the zone of molecular attraction, the water cannot
move. Thus, coarse-grained rocks are usually more permeable than fine-grained rocks, and
sands are more permeable than clays.
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Factor Affecting the Magnitude of Permeability:
i) Shape and size of sand grains: Rock composed of large and flat grain uniformly
arranged with longest dimension horizontal then the permeability will be vary than the
vertical permeability.
ii) Lamination: Lamination of shale and platy minerals such as muscovite act as barrier to
vertical permeability.
iii) Cementation: Excess of cement in the pore space reduces the permeability.
Intrinsic Permeability: The intrinsic permeability represents the physical flow properties of
the geologic materials. It is a more rational concept as it is independent of fluid properties
and depends on the properties of the medium. The larger the pore opening, the larger the
intrinsic permeability of the medium.
Hydraulic Conductivity: The hydraulic conductivity of a soil is a measure of the soil's
ability to transmit water when submitted to a hydraulic gradient. It depends on the soil
grain size, the structure of the soil matrix, the type of soil fluid, and the relative amount of
soil fluid (saturation) present in the soil matrix. For a subsurface system saturated with the
soil fluid, the hydraulic conductivity, K , can be expressed as follows (Bear 1972)
k gK
(1.3)
where k is the intrinsic permeability of the soil, is the fluid density, is the fluid
viscosity and g is the force due to gravity.
1.8 Darcy’s Law:
In the mid-nineteenth century, Henry Darcy (1856) systematically studied the
movement of water through sand columns. He was demonstrated from his experiment that
the rate of flow, i.e., volume of water per unit time, Q is directly proportional to the cross-
sectional area, A , and head loss, h and inversely proportional to the length of the
flow path, L shown in Fig. 1.9.
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Fig. 1.9 Flow of water through an inclined porous media
hQ A
L
Darcy’s law can be written as
hQ KAL
(1.4)
where K Hydraulic conductivity of the porous medium and –ve sign indicates
that Q occurs in the direction of the decreasing head.
Here, h hL l
Hydraulic gradient.
Darcy’s law can also be written as:
Q hq KA l
(1.5)
where q The Darcian velocity, also known as the specific discharge. Eq. (1.5) simply
denotes that the specific discharge is the volume of water flow per unit time through a unit
cross-sectional area normal to the direction of flow.
Validity of Darcy’s Law: Darcy's law was established in certain circumstances: laminar
flow in saturated granular media, under steady-state flow conditions, considering the fluid
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homogenous, isotherm and incompressible, and neglecting the kinetic energy. Still, due to
its averaging character based on the representative continuum and the small influence of
other factors, the macroscopic law of Darcy can be used for many situations that do not
correspond to these basic assumptions (Freeze and Cherry, 1979)
saturated flow and unsaturated flow;
steady-state flow and transient flow;
flow in granular media and in fractured rocks;
flow in aquifers and flow in aquitards;
flow in homogeneous systems and flow in heterogeneous systems;
flow in isotropic media and flow in anisotropic media.
The most restrictive hypothesis of Darcy's law is one that consider the laminar flow,
and the fluid movement as dominated by viscous forces. This occurs when the fluids are
moving slowly, and the water molecules move along parallel streamlines. When the
velocity of flow increases (for instance in the vicinity of a pumping well), the water
particles move chaotically and the streamlines are no longer parallel. Therefore the flow is
turbulent, and in that case the inertial forces are more influential than the viscous forces
(Fetter, 2001).
The ratio between the inertial forces and the viscous forces driving the flow is computed
by the Reynolds number, which is used as a criterion to distinguish between the laminar
flow, the turbulent flow and the transition zone. For porous media, the Reynolds number is
defined as:
vdRe
(1.6)
where is the fluid density, v is the velocity, d the diameter (of a pore), and the
viscosity of the fluid. According to Bear (1972), Darcy' law with consideration of a laminar
flow is valid for Reynolds number less than 1, but the upper limit can be extended up to 10
shown in Fig. 1.10.
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Fig. 1.10 Range of validity of Darcy's law
The inception of the turbulent flow can be located at Reynolds numbers greater than
60…100. Between the laminar and the turbulent flow there is a transition zone, where the
flow is laminar but non-linear. In a general way, Darcy's law can be written: nhq K
L
(1.7)
When 1n , the law is linear and the flow is laminar which is the case of Darcy’s law
given in Eq. (1.5). As per the Fig. 1.10, the hydraulic conductivity is the slope of the
straight line in the interval of validity of Darcy's law. When 1n and 100Re , the law is
non-linear and the flow is turbulent.
1.9 Contaminant Transport Mechanism
Contaminant transport mechanisms usually concerned with movement in the saturated
zone, however, in many cases the unsaturated zone may not be ignored. The details of
contaminant transport mechanism are discussed below:
Advection: - Advection (convection) is the movement of solute caused by the groundwater
flow. The bulk movement of water through the aquifer causes solute transport via
advection. Advection is the primary process by which solute moves in the groundwater
system. Due to advection, non-reactive solute travel at an average rate equal to the seepage
velocity of the fluid,
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dvu
(1.8)
where u is the seepage or pore water velocity of the groundwater, is the porosity of the
porous material, and dv is the flux of water (i.e., quantity of water per unit area per unit
time).
Diffusion: Diffusion is the transport process in which the chemical species migrate in
response to a gradient in its concentration. This is a physical phenomena linked to the
molecular agiation, and occurs in a fluid even at rest due to Brownian motion of particles.
The result of this molecular agiation is that the particles are transferred from zones of
higher concentration to those of lower concentration. A hydraulic gradient is not required
for transport of contaminant by diffusion. The fundamental equation for diffusion is Fick’s
first law which is, for one dimensional in free solution (i.e., no porous media), can be
written as,
0DcJ Dx
(1.9)
where DJ is the diffusive mass flux, x is the direction of transport and 0D is the ‘free
solution’ diffusion coefficient. For diffusion in saturated porous material, a modified form
of Fick’s first law is used,
0D
cJ Dx
(1.10)
or DcJ Dx
(1.11)
where is the dimensionless tortuosity factor and D is the effective diffusion coefficient.
Dispersion: Dispersion is the spreading of the plume that occurs along and across the main
flow direction due to aquifer heterogeneities at both the small scale (pore scale) and at the
macroscale (regional scale). Dispersion tends to increase the plume uniformity as it travels
downstream. Factors that contribute to dispersion include: faster flow at the center of the
pores than at the edges, some pathways are longer than others, the flow velocity is larger in
smaller pores than in larger ones. This is known as mechanical dispersion.
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Hydrodynamic dispersion is the result of two processes, molecular diffusion and
mechanical mixing or mechanical dispersion. Molecular diffusion is the process whereby
ionic or molecular constituents move under the influence of their kinetic activity in the
direction of their concentration gradients. Under this process, constituents move from
regions of higher concentration to regions of lower concentration; the greater the
difference, the greater the diffusion rate (Yong and Ahmed, 1999).
Dispersion in porous material refers to the spreading of a stream or discrete volume of
contaminants as it flows through the subsurface. For example, if a spot of dye is injected
into porous material through which groundwater is flowing, the spot is enlarge in size as it
moves down-gradient. Dispersion causes mixing with uncontaminated groundwater, and
hence dispersion is a mechanism for dilution. Moreover, dispersion causes the contaminant
to spread over a greater volume of aquifer than would be predicted solely from an analysis
of groundwater velocity vectors. This spreading effect is of particular concern when toxic
or hazardous wastes are involved. Dispersion is primarily importance in predicting
transport away from point sources of contamination but is also influential in the spread of
non point-source contaminants, although of lesser importance. Contaminants introduced
into the subsurface from non-point sources spreads over a relatively large area because of
the nature of the loading pattern. In this case, dispersion merely causes a relatively large
zone of contaminated water to acquire some rough fringes. Dispersion is of interest because
it causes contaminants to arrive at a discharge point (e.g., a stream or a water well) prior to
the arrival time calculated from the average groundwater velocity. The accelerated arrival
of contaminants at a discharge point is a characteristic feature of dispersion that is due to
the fact that some parts of the contaminant plume move faster than the average
groundwater velocity (Yong and Ahmed, 1999).
Sorption: Sorption refers to the exchange to molecules and ions between the solid phase
and the liquid phase. It includes adsorption and desorption. Adsorption is the attachment of
molecules and ions from the solute to the rock material. Adsorption produces a decrease of
concentration of the solute or equivalently, causes a retardation of the contaminant
transport compared to the water movement. Desorption is the release of molecules and ions
from the solid phase to the solute.
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1.10 Advection-Dispersion Equation
Hydrodynamic dispersion of solute through a porous medium is described by a
partial differential equation known as advection-dispersion equation. This partial
differential equation is of parabolic type (Guenther and Lee, 1988; Logan, 1994).
Analytical solutions to advection-dispersion equations are of continuous interest because
they present benchmark solutions to the problems in the assessment of degradation of
hydro-environment (Bear and Verruijit, 1987).
Let a solute be in the moving liquid which is entertained by the flow. Let us
consider a small cubical element of volume dxdydz of sides PQ dx , PS dy , PA dz ,
surrounding a position P ( , , )x y z in a Cartesian three dimensional frame of reference as
shown below in Fig. 1.11.
Figure 1.11 Small cubical element
Let the concentration at this position be ( , , )C x y z . Mass entering the element
through the face PADS is xJ dydz . Mass leaving the element through the face QBCR is
( )x dxJ dydz . Net gain inside the element along x -axis
( ) ....x x dx x x xdydz J J dydz J J dxJ xJdxdydzx
(1.12)
C D
A B
Q P
R S
xJ dydz
( )x dxJ dydz
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Similarly net gain inside the element along y -axis and z -axis can be obtained as
yJdxdydz
y
, and zJdxdydz
z
, respectively. Total gain inside the element can now
be written as
yx zJJ Jdxdydzx y z
By Fick’s law of diffusion, we have
xCJx
, y
CJy
and z
CJz
(1.13)
where ve sign in first proportionality occurs because the x -axis direction is from higher
concentration to lower concentration, and so on. The diffusive current densities
.( , , )diff x y zJ J J J
and convective current density .convJ
through the element are given as
follows (Nield and Began, 1992)
x xCJ Dx
; y y
CJ Dy
; z z
CJ Dz
(1.14)
and .convJ uC (1.15)
where xD , yD , zD are dispersion coefficients along x -axis, y -axis and z -axis,
respectively and ( , , )x y zu u u u is the flow velocity. Total current density
1 2 3( , , )J J J J
through the element can be written as
1 2 3 . .( , , ) diff convJ J J J J J
(1.16)
According to the conservation of solute mass inside the volume element, we have
31 2 JJ JCdxdydz dxdydzt x y z
(1.17)
or 31 2 0JJ JCt x y z
(1.18)
Using relationships (1.14-1.16), Eq. (1.18) becomes
( ) ( ) ( )x y z x y zC C C CD D D u C u C u Ct x x y y z z x y z
(1.19)
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This equation is known as advection-dispersion equation in general form in three
dimensions subject to the condtion that one of the axis coincides with direction of average
velocity. In principle, the coefficients xD , yD , and zD may be function of position, time as
well as concentration. If it is not then these components are called dispersion coefficients.
In case all the coefficients in Eq.(1.19) are constants, the advection-dispersion
equation becomes 2 2 2
2 2 2x y z x y zC C C C C C CD D D u u ut x y z x y z
(1.20)
The advection-dispersion Eq. (1.20) in one dimension along x -axis direction in general
form becomes 2
2x xC C CD ut x x
(1.21)
1.10 Initial and Boundary Conditions:
Initial Condition: For transport problems, the initial conditions are represented by the
extent and concentrations of an existent contaminant plume at the starting time of
simulation. The measured concentration distribution of a plume may not be used as starting
point for a transient transport simulation due to the scarce information. The monitoring
program usually does not predict a large number of measuring points and frequently the
highest-occurring concentrations are not even registered. A solution is to consider
estimated source terms as starting point for the transport model.
Boundary Conditions: The different types of boundary conditions are described as follows:
i) First kind or Dirichlet condition: It is also named as concentration-type condition or
concentration boundary is used to represent known recharge with known concentration (the
advective flux) at the input end of the model. Usually, the transport model area is chosen
large enough so that the contaminants do not reach the outflow end. In this way one avoids
introducing unknown concentrations at the outflow boundary; at the same time, the
dispersive flux at the boundaries is set to zero (Spitz and Moreno, 1996). Structures like
leaking landfills, ponds, infiltration beds, injections wells can be modeled by a prescribed
flux with associated concentration. The same conditions can be used for water losses from
irrigation systems, sewage systems or industrial areas. To model a source of pure
23
contaminant (oil, mine waste) prescribed concentration can be imposed inside the model
area. Surface bodies serving as output for contaminated groundwater can be modeled by
using a prescribed head and concentration boundary; in this case, the value of the
concentration corresponds to the concentration of the surface water body.
ii) Second kind or Neumann condition: It is specifying the concentration gradient
prescribes only dispersive flux, meaning it neither prescribes total mass flux (advective and
dispersive flux combined), nor the advective flux alone (Spitz and Moreno, 1996). As
shown before, at the boundaries the dispersive flux is set to zero. The same value is
considered along the impervious boundaries.
iii) Third kind, mixed or Cauchy condition: It prescribes the total contaminant flux on
the boundary as a linear combination of the concentration (advective flux) and
concentration gradient (dispersive flux). For large periods of time, due to mixing across the
boundary the dispersive flux diminishes and the third kind condition reduces to first kind
condition. When the advective flux tends to zero, the third kind condition is approaching
the second type condition.
1.12 Literature Survey
Subsurface water is generally divided into two major types: phreatic water or soil
moisture in the unsaturated zone and groundwater in the saturated zone. Groundwater is a
long-term reservoir of the natural water cycle, which originates from rainfall or snow. The
groundwater resources are being utilized for drinking, irrigation and industrial purposes.
Groundwater resources also play a major role in ensuring livelihood security across the
world, especially in economies that depends on agriculture. India is now the biggest user of
groundwater for agriculture in the world (Shah, 2009). Groundwater irrigation has been
expanding at a very rapid pace in India since the 1970s. The data from the Minor Irrigation
Census conducted in 2001 shows evidence of the growing numbers of groundwater
irrigation structures (wells and tube wells) in the country (Shankar et al., 2011).
The natural and generally high quality groundwater is under attack today by many
sources and types of contaminants which are associated with human activities and land use
(Gelhar and Wilson, 1974). Therefore, groundwater systems, planning, and management
are needed for judicious use of groundwater. In recent years, an increasing threat to
24
ground- water quality due to human activities has become of great importance. The adverse
effects on groundwater quality are the results of man's activity at ground surface,
unintentionally by agriculture, domestic and industrial effluents, unexpectedly by sub-
surface or surface disposal of sewage and industrial wastes. The intensive use of natural
resources and the large production of wastes in modern society often pose a threat to
groundwater quality and have already resulted in many incidents of ground water pollution.
Pollutants are being added to the groundwater system through human activities and natural
processes. The groundwater pollution normally traced back to four main sources such as
industrial, domestic, agricultural and environmental pollution by Sharma and Reddy
(2004). Solid waste from industrial units is being dumped near the factories, which is
subjected to reaction with percolating rain water and reaches the ground water level.
Groundwater pollution caused by human activities usually falls into one of two categories:
point-source pollution and non point-source pollution. Simultaneous movement of solute
and water occurs in every imaginable natural flow system in soil (Waric et al., 1971). This
phenomenon occurs during leaching of solutes, movement of fertilizers and pesticides, and
land reclamation activities (Bresler and Hanks, 1969). Fertilizers and pesticides applied to
crops finally may reach underlying aquifers, particularly if the aquifer is shallow and not
"protected" by an overlying layer of low permeability material, such as clay. After
infiltration through soil these pollutants reach the groundwater table. As soon as the
pollutant reaches the groundwater table, it starts spreading along the groundwater flow in
horizontal directions. Drinking-water wells located close to cropland sometimes are
polluted by these agricultural chemicals. The percolating water picks up a large amount of
dissolved constituents and reaches the aquifer system and contaminates the groundwater.
The problem of groundwater pollution in several parts of the country has become so acute
that unless urgent steps for detailed identification and abatement are taken, extensive
groundwater resources may be damaged. The concentration of pollutants decreases due to
the hydrodynamic dispersion and other attenuation effects, like adsorption, first order decay
terms, etc. Coal mines are another major source of contaminants. When pyrite rocks
associated with coal mining are exposed to oxygen they are oxidized to generate acid mine
drainage. The waste then flows into streams and infiltrates into aquifers. A vast majority of
25
groundwater quality problems are caused by contamination, over-exploitation, or
combination of these two. Most groundwater quality problems are difficult to detect and
hard to resolve. The solutions are usually very expensive, time consuming and not always
effective. Groundwater quality is slowly but surely declining everywhere. The wide range
of contamination sources is one of the many factors contributing to the complexity of
groundwater assessment.
Groundwater Contamination is more complex than surface water mainly because of
difficulty in its timely detection and slow movement. The cleanup of contaminated
groundwater is very different from clean up of waste-water on the surface. The most
obvious difference is that with groundwater clean-ups, the water body is actually being
cleaned. In surface water cleanups, we control and treat the waste water that is entering a
water body. The water body, river or lake actually cleans itself once we stop putting
pollutants into it. Groundwater is not able to clean itself; hence we must clean-up the
source of pollutants as well as the aquifer itself. With growing recognition of underground
water resources efforts are increasing to prevent, reduce and eliminate groundwater
contamination by the researchers in various discipline. The researchers and scientists from
various disciplines like, Hydrogeology, hydrology, civil engineering, environmental
science engineering etc. putting their best effort to solve this problem by various means.
Mathematical modeling is one of the powerful tools to project the existing problems and its
appropriate solution. These models are based on certain simplifying assumptions, have
been used to predict groundwater flow and solute transport. A solute transport model is an
essential tool for assessing environmental risks to groundwater resources. These processes
are described by a partial differential equation (PDE) of parabolic type, and it is usually
known as advective-dispersive equation (ADE). This transport equation is one of the
fundamental equations in hydrodynamics and plays a significant role in water quality and
solute transport modeling. A solute transport model have been solved analytically for
simple cases and numerically for more complicated systems. Many solute transport
problems have been solved numerically (Rastogi, 2007) but analytical solutions are still
pursued by many scientists, because they can provide physical insights into problems
(Batu, 2006).
26
Analytical solutions for solute transport equation were developed by many
researchers (Gershon and Nir, 1969; Gelher and Collins, 1971; Marino, 1974; van
Genuchten, 1981; Hunt, 1983) to describe one dimensional solute transport. Kumar (1983a)
presented analytical solutions for dispersion (in a finite non-adsorbing and adsorbing
porous media) which was controlled by flow (with unsteady unidirectional velocity
distribution) of low concentration fluids towards a region of higher concentration. van
Genuchten and Wierenga (1986) compiled analytical solutions of the advective-dispersive
equation (ADE) for different boundary conditions in finite and semi-infinite domains.
Lindstrom and Boersma (1989) obtained an analytical solution of the general one-
dimensional solute transport model for confined aquifers. Yates (1990) developed an
analytical solution for describing the transport of dissolved substances in heterogeneous
porous media with a distance-dependent dispersion relationship. An analytical solution for
one-dimensional dispersion in unsteady flow in an adsorbing porous medium was obtained
by Yadav et al. (1990). Fry and Istok (1993) derived an analytical solution for the
advection-dispersion equations with rate limited desorption and first-order decay, using an
eigenfunction integral equations method. An analytical solution of the one-dimensional
time-dependent transport equation was also discussed by Basha and Habel (1993). Sim and
Chrysikopoulos (1996) were presented analytical solutions of one dimensional virus
transport in saturated, homogeneous porous media with time-dependent inactivation rate
coefficients. Chrysikopoulos and Sim (1996) were also developed a stochastic model for
one-dimensional virus transport inhomogeneous, saturated, semi-infinite porous media.
Logan (1996) presented an analytical model of solute transport in porous media with scale
dependent dispersion and periodic boundary conditions. An analytical solution for
contaminant transport in non-uniform flow was presented by Tartakovsky and Federico
(1997). Kumar and Kumar (1998) presented analytical and numerical solutions for
homogeneous and inhomogeneous semi infinite aquifers respectively. The water table
variation in response to time varying recharge was explored by Rai and Manglik (1999).
Analytical solutions for solute transport in saturated porous media with semi-infinite or
finite thickness were discussed by Sim and Chrysikopoulos (1999). Virus transport in
unsaturated porous media was also discussed by Sim and Chrysikopoulos (2000). Kumar
27
and Yadav (2000) have discussed a solute dispersion along and against sinusoidally varying
unsteady velocity through finite aquifers Analytical/Numerical solutions. A one-
dimensional transport model for simulating water flow and solute transport in
homogeneous–heterogeneous, saturated–unsaturated porous media using the discontinuous
finite elements method was presented by Diaw et al., (2001). Didierjean et al., (2004) was
discussed some analytical solutions of one-dimensional macro-dispersion in stratified
porous media by the quadrupole method with different types of heterogeneous porous
medium. An analytical solution to transient, unsaturated transport of water and
contaminants through horizontal porous media was discussed by Sander and Braddock
(2005). Recently some more investigations have been carried out for one-dimensional
solute transport in saturated or unsaturated porous media using different mathematical
approaches. Analytical solutions for sequentially coupled one-dimensional reactive
transport problems were discussed by Srinivasan and Clement (2008). Longitudinal
dispersion with time dependent source concentration along unsteady groundwater flow in
semi-infinite aquifer was presented by Singh et al., (2008). An analytical solution for one-
dimensional contaminant diffusion through multi-layered media is derived regarding the
change of the concentration of contaminants at the top boundary with time by Chen et al.,
(2009). Taylor-Galerkin B-spline finite element method for the one-dimensional advection-
diffusion equation was also discussed by Kadalbajoo and Arora (2009). Jaiswal et al.,
(2009) presented an analytical solution for temporally and spatially dependent solute
dispersion of pulse type solute concentration in one-dimensional semi-infinite media.
Solute transport for one-dimensional homogeneous porous formations with time dependent
point-source concentration was presented by Singh et al., (2009). Singh et al., (2010b) also
presented an analytical solution for solute transport along and against time dependent
source concentration in homogeneous finite aquifers. P´erez Guerrero and Skaggs (2010)
were derived an analytical solution for one-dimensional advection-dispersion transport
equation with distance dependent coefficients in heterogeneous porous media. Kumar et al.,
(2010) were derived analytical solutions for one dimensional advection-diffusion equation
with variable coefficients in steady flow through an inhomogeneous medium, temporally
dependent solute dispersion along uniform flow through homogeneous and temporally
28
dependent flow through inhomogeneous medium. Analytical solution for contamination
diffusion in doubled layered porous media was also discussed by Li and Cleall (2010). An
analytical solution of the convection–dispersion–reaction equation is obtained for a finite
one-dimensional region with a pulse boundary condition was presented by Ziskind et al.,
(2011). Qiu et al., (2011) were explored a transient storage model with space-variable
coefficients and solved using the generalized integral transform technique (GITT) coupled
with Laplace transform method. Analytical solutions of one-dimensional advection-
diffusion equations (ADE) subject to an initially pollutant-free domain and varying pulse-
type input conditions using Laplace integral transform technique (LITT) in semi-infinite
heterogeneous medium presented by Singh et al., (2012).
Multidimensional solute transport problems in saturated media have been attracting
the attention of many researchers. Two-dimensional solute transport problems involve both
longitudinal as well as transverse dispersion along with porous media flow in addition to
advection. Bruce and Street (1967) presented both longitudinal and lateral dispersion in
semi-infinite non-adsorbing porous medium in a steady flow field for a constant input
concentration. van Genuchten et al., (1977) were discussed the simulation of two-
dimensional contaminant transport with isoparametric hermitian finite element. Hunt (1978
a, b) presented one, two and three-dimensional solutions for instantaneous, continuous, and
steady-state pollution sources in uniform groundwater flow. Latinopoulos et al., (1988)
presented a method to obtain the analytical solutions for chemical transport in two-
dimensional aquifers in which a constant velocity field, linear adsorption, and first-order
decay were considered. A generalized two-dimensional analytical solution for
hydrodynamic dispersion in bounded media with the first-type boundary condition at the
source was discussed by Batu (1989). Batu and van Genuchten (1990) were discussed first
and third-type boundary conditions in two-dimensional solute transport modeling. A
generalized two-dimensional analytical solute transport model in bounded media for flux
type finite multiple sources was discussed by Batu (1993). A mathematical modeling for
transient solute transport resulting from the dissolution of a single component nonaqueous
phase liquid (NAPL) in two-dimensional, saturated, homogeneous porous media were
presented by Chrysikopoulos et al., (1994). Analytical solution for two dimensional
29
transport equations with time dependent dispersion coefficients was derived by Aral and
Liao (1996). Tartakovasky (2000) presented an analytical solution for two-dimensional
contaminant transport during groundwater extraction. Vogel et al., (2000) also discussed
the solute transport in a two-dimensional dual-permeability system with spatially variable
hydraulic properties in heterogeneous soil system. Two dimensional analytical solution,
using Hankel transformation, was presented by Kumar and Kumar (2002). Chen et al.,
(2003) discussed a Laplace-transformed power series (LTPS) technique to solve a two-
dimensional ADE in cylindrical co-ordinate with non-axisymmetry solute transport in a
radially convergent flow field. Some analytical solutions for two-dimensional convection-
dispersion equation in cylindrical geometry with chemical decay or adsorption-like reaction
inside the liquid phase were discussed by Massabo et al., (2006). Chen (2007) was
presented a two-dimensional power series solution for non-axisymmetrical transport in a
radially convergent tracer test with scale dependent dispersion. James and Jawitz (2007)
were also discussed a two-dimensional reactive transport ADE model using a splitting
technique where advective, dispersive and reactive parts of the equation were solved
separately. A two-dimensional contaminant transport through saturated porous media using
a mesh free method called the radial point interpolation method (RPIM) with polynomial
reproduction, was derived by Kumar and Dodagourdar (2008). Zhan et al., (2009)
presented an analytical solution of two-dimensional solute transport in an aquifer-aquitard
system by maintaining rigorous mass conservation at the aquifer-aquitard interface with the
first-type and the third type boundary conditions. An Analytical Solution for the Transient
Two-Dimensional Advection–Diffusion Equation with Non-Fickian Closure in Cartesian
Geometry by the Generalized Integral Transform Technique (GILTT) were derived by
Buske et al., (2010). Singh et al., (2010a) were derived an analytical solution using Hankel
Transform Technique (HTT) for two-dimensional solute transport in finite aquifer with
time dependent source concentration. Lin et al., (2011) were presented an analytical
solution of two-dimensional ADE in cylindrical geometry, finite length medium with third-
type inlet boundary conditions has been discussed using the second kind finite HTT and the
generalized integral transform technique (GITT) and explored the fact that the influence of
exit boundary conditions diminishes, when Peclet number increases. Analytical solutions
30
for two-dimensional ADE in cylindrical coordinate subject to the third- type inlet boundary
condition were presented by Chen et al., (2011a). Chen et al., (2011b) were also discussed
the exact analytical solutions for two-dimensional ADE in cylindrical coordinates subject
to finite exit boundary. Zhang (2011) was discussed a two-dimensional simulation study on
longitudinal solute transport and longitudinal dispersion coefficient. Jaiswal et al., (2011)
were presented analytical solutions obtained for two-dimensional ADE describing the
dispersion of pulse-type point source along temporally and spatially dependent flow
domain respectively, through a semi-infinite horizontal isotropic medium. Yadav et al.,
(2012) were derived an analytical solution for horizontal solute transport from a pulse type
source along temporally and spatially dependent flow.
Three-dimensional analytical models for solute transport in finite and semi-infinite porous
media were discussed by Goltz and Roberts (1986), Yates (1988), Leij et al., (1991, 1993),
Chrysikopoulos (1995) etc. A generalized three-dimensional analytical solute transport
model for multiple rectangular first-type sources was discussed by Batu (1996). Sim and
Chrysikopoulos (1998) were presented three-dimensional analytical solutions for solute
transport in saturated, homogeneous porous media with semi-infinite or finite thickness.
Zoppou and Knight (1999) derived an analytical solution for the two- and three-
dimensional ADE with spatially variable velocity and diffusion coefficients. Park and Zhan
(2001) discussed analytical solutions of contaminant transport from finite one-, two-, and
three-dimensional sources in a finite thickness aquifer using Green’s function method. An
analytical solution of the advection-diffusion transport equation using a change-of-variable
and integral transform technique were presented by P´erez Guerrero et al., (2009).
1.13 Numerical Input Data To compute the analytical solutions of the problems present in this thesis, the input
data of initial groundwater velocity and dispersion coefficient are required. The input
values are chosen on the basis of inputs available in the previous works. Some of the works
and values of these two parameters are enlisted below in the form of table.
31
Author Year Seepage velocity ( u )
Dispersion ( D )
Al-Niami and Ruston
1979 35 cm2/day and 8x10-4 cm/day
0.1 cm2/sec and 0.0001 cm2/sec
Marino 1981 0.005 ft/min 0.1 ft2/min van Genucheten and Alves
