IIT BHU training

8/12/2019 IIT BHU training

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1. INTRODUCTION1.1GROUND WATER

A major component of precipitation that falls on the earth surface eventually enters into the

ground by the process of infiltration. The infiltrated water is stored in the pores of theunderground soil strata. The water which is stored in the pores of the soil strata is known as

groundwater. Therefore, the groundwater may be defined as all the water present below the

earth surface and the groundwater hydrology is defined as the science of occurrence,

distribution and movement of water below the earth surface.

Water falls from the atmosphere to the land in the form of precipitation (rain, snow, sleet, or

hail). Once on land, some of the precipitation may accumulate as surface water in streams

and rivers, lakes, ponds, reservoirs, wetlands, and oceans. Some water may seep downward

through the soil where it is stored as groundwater. Here the spaces between sand and gravel

deposits or cracks in the bedrock are filled with water.

Figure 1.1The hydrologic cycle

Once on land, the water may: (1) evaporate from land and re-enter the atmosphere directly,

(2) flow into rivers and streams, or (3) seep downward in the soil and become groundwater.

Every molecule of water is moving through the hydrologic cycle.

Groundwater begins as precipitation and snowmelt that seeps, or infiltrates into the ground.

The amount of water that infiltrates into the ground varies widely from place to place depending

on the type of land surface and soil present. Water readily seeps into permeable material, like

soil. However, water runs off impermeable areas like paved driveways, parking lots, roads,

and compacted soil. The rest of the precipitation and snowmelt that does not seep into the

ground runs off the land surface into water bodies or returns to the atmosphere by evaporation.


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1.2STORAGE OF GROUND WATERAn aquifer is an underground geological formation which contains water and sufficient

amount of water can be extracted economically using water wells. Aquifers comprise generally

layers of sand and gravel and fracture bedrock.When water table serves as the upper boundaryof the aquifer, the aquifer is known as unconfined aquifer. As discussed in the earlier section,

there exists a capillary zone above the water table. An aquifer which is bounded by two

impervious layers at top and bottom of the aquifer is called confined aquifer. In case of

confined aquifer, if we insert a piezometer into the aquifer, the water level will rise above the

top impervious layer as the pressure in the aquifer is more than the atmospheric pressure.

Figure 1.2Confined and Unconfined Aquifer

Permeability and porosityare important when considering the amount of water an aquifercan hold and supply to a well. Permeability is a measure of how readily water flows through

connected openings in soil. An impermeable material does not allow water to pass through it.

Sands and gravels have a high permeability as compared to dense bedrock or a clay material.

Porosity is the capacity of soil or rock to hold water. Most watermovement occurs through thelarger soil pores. Although sands and gravels have a lower porosity as compared to clays, they

do have larger pores.


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1.3 DISCHARGE AND RECHARGE OF GROUNDWATER

Groundwater recharge and discharge are important although typically inconspicuous aspectsof the global hydrological cycle. Rechargeis the replenishment of groundwater by downward

infiltration of precipitation, or by water that was temporarily stored on the earths surface

Recharge involves the downward movement and influx of groundwater to an aquifer.

Discharge involves the upward movement and out flux of groundwater to an aquifer. Recharge

and Discharge activities are usually limited to a small portion of an aquifer.

1.3.2 Discharge and Recharge in Unconfined Aquifer

Groundwater can directly infiltrate an unconfined aquifer from Earths surface and gravitate

towards the water table, hence recharge represents a net downward vertical influx of water

to the water table. Recharge area are most rigorously defined as those regions in which there

exist a downward component to the groundwater flow lines and this flow is directed away from

the water table. In order to gravitate soil pores must become mainly filled with water or reach

field capacity. This is necessary in order to overcome the often high tension or capillary force

within the vadose zone. Because of these recharge consideration recharge is intermittent, rather

than a continuous process that is limited to periods of excessive rainfall that leave the soil wet.

Most of the Recharge areaslocated below topographically raised areas, making the water tablea profile a rubbed image of the surface topography. Multiple Groundwater flow path can be

taken from a recharge area, including short and shallow paths to locate tributaries and longer

deeper paths to regional stream basins

Figure 1.3 Recharge of Unconfined Aquifer


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1.3.2 Recharge and Discharge in Confined Aquifers

Recharge and Discharge process are less amenable to direct observation or study in confined

aquifer than in unconfined aquifer. The most common type of recharge area associated with

the confined aquifers is at an outcrops area where the aquifer emerges upon the earths surface

either at an updip location or where the erosion has removed part of overlaying confining units.

Recharge to confined aquifer also occurs through basses of sinkholes, lakes and storage

reservoir providing that the bottom of these features are in contact with the confined aquifer

and that the hydraulic head associated with these features is greater than that associated with

sinkholes, lakes and the reservoirs so the latter features become loci for Groundwater discharge.

Groundwater can both enter and leave a confined aquifer deep below the Earths surface,

passing through openings in the overlaying or underlying confining beds. Is the Hydraulic head

is greater in the confined aquifer that in the overlaying or underlying aquifers, groundwater can

leak or flow through the confining layer to these aquifers. Conversely, if the hydraulic head of

an overlaying or underlying aquifer is greater than the head within the confined aquifer, waterfrom those aquifers can then recharge the confined aquifer.

Figure 1.4Discharge of a Confined Aquifer

Figure 1.5 Recharge of Confined Aquifer


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2. GROUNDWATER MODELLING

2.1 OBJECTIVE OF GROUNDWATER MODELLING

Groundwater models provide a tool to estimate groundwater availability for various water use

strategies and to determine the cumulative effects of increased water use and drought. A

groundwater model is a numerical representation of the aquifer system capable of simulatinghistorical conditions and predicting future aquifer conditions. Inherent to the groundwater

model are a set of equations that are developed and applied to describe the primary or dominant

physical processes considered to be controlling groundwater flow in the aquifer system.

Groundwater models are essential to performing complex analyses and in making informed

predictions and related decisions.

The mathematical modelling of regional aquifers has the following purposes:

1. To predict the future behaviour of the studied aquifer in response to stress-factors likenew pumping, infiltration changes due to global changes, irrigation, and pollutantcontaminations.

2. To provide information required in order to comply with local regulations.3. To obtain a better understanding of the aquifer system from the geological,

hydrogeological and hydro chemical point of view.

4. To provide information for the improving of the observation networks and fieldexperiments.

5. To determine the interactionof Groundwater and Surface water.6. To determine the discharge and recharge ratesof the Groundwater.

The geologic medium is not an engineering system, where all the characteristics or parameters

are deterministically known. The parameters expressing the characteristics of the aquifer

(hydraulic conductivity, effective porosity, dispersivities etc.) are equivalent or averaged

values. The hydrogeological parameters are obtained through a calibration process applied for

the whole aquifer, based on the best agreement between the measured and the computed levels.

Local heterogeneities are taken into account by different values of the chosen parameters. The

accuracy of the simulation strongly depends on the scale of the problem.

It has been recently shown that contaminant transport properties of geological media cannot be

deduced from laboratory tests. A strong scale effect affects, among others, the dispersivitycoefficients. A reliable and quantitative way of determining the aquifer transport properties

consists in interpreting two-well injection-pumping tracer tests. A pulse of tracer-labeled water

is injected in one well and pumping in a nearby well creates a radially converging flow field.

Tracer breakthrough curves are measured in this last well. This information is used to evaluate

aquifer transport properties and to calibrate the contaminant transport model. Interpretation by

modelling local transport conditions of the aquifer is not an easy task.


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For assessing protection zones around main pumping wells, detailed simulations of

groundwater transport behaviour must be completed involving:

the local calibration of the model for flow based on measured piezometric levels ; the local calibration for transport (on the measured breakthrough curves from tracer

tests or measured contaminations) ;

the simulations and computations of travel times for different injection points.

The Groundwater availability model was developed using a modelling protocol that is standard

to the groundwater modelling industry. This protocol includes:

(1) the development of a conceptual model for groundwater flow in the aquifer, including

defining physical limits and properties,

(2) model design,

(3) model calibration,

(4) sensitivity analysis, and

(5) reporting.

The conceptual model is a conceptual description of the physical processes governing

groundwater flow in the aquifer system. Available data and reports for the model area werereviewed in the conceptual model development stage. Model design is the process used to

translate the conceptual model into a physical model, in this case a numerical model of

groundwater flow. This involves organizing and distributing model parameters, developing a

model grid and model boundary conditions, and determining the model integration time scale.

Model calibration is the process of modifying model parameters so that observed field

measurements (e.g., water levels in wells) can be reproduced. The model was calibrated to

steady-state conditions representing, as closely as possible, conditions in the aquifer prior to

significant development and to transient aquifer conditions. Sensitivity analyses were

performed on both the steady-state and transient portions of the model to offer insight to the

uniqueness of the model and the impact of uncertainty in model parameter estimates.

2.2 INPUTS FOR GROUNWATER MODELLING

For the calculations one needs inputs like:

hydrological inputs, operational inputs, external conditions: initial and boundary conditions, (Hydraulic) parameters.The model may have chemical components like water salinity, soil salinity and other qualityindicators of water and soil, for which inputs may also be needed.


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Input data that is required for code simulations can be classified into:

1.Geometry and topography issues(a) Site boundaries and dimensions

(b) Surface topography (e.g. to detect zones with surface infiltration)

(c) Location of streams, divides, ponds and so on (d) Land use (landfills, dikes, well locations,

irrigation)

2. Geology and hydrology issues

a) Aquifers (stratifi

cation, depth, lithological parameters, hydraulic conductivities, longitudinaland transversal dispersivities, storativities (i.e. matrix and water compressibilitys), porosities)

(b) Porous medium density

(c) Water levels at surface reservoirs (rivers, ponds,etc.) compressing shallow aquifers(d) Pumping/recharge point sources (well depth,intensity, periodicity, and time of application)(e) Distributed sources of inflow, for example, rain-fall and irrigation rates

(f) Distributed sources of outflow, for example, evapotranspiration

(g) Time dependent data at spatial points

3. Water and porous medium chemical properties

(a) Sorption (adsorption and desorption) factors.

(b) Electrical conductivities.

(c) Temporal and spatial concentration of solutes in the water and the solid phases of the

porous medium

(d) Solutes associated with sources of recharge fluxes.

e) Concentration of stable isotopes and microelements.

4. Boundary and initial conditions

(a) Initial field distribution of piezometric head and components concentration.

(b) Pervious/impervious boundary segments with the ascribed flux conditions.

(c) Piezometric heads and concentrations along boundaries.


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2.3 DIMENSIONS

Groundwater models can be one-dimensional, two-dimensional, three-dimensional and semi-

three-dimensional. Two and three-dimensional models can take into account the anisotropy of

the aquifer with respect to the hydraulic conductivity, i.e. this property may vary in different

directionsOne, Two and Three-dimensional

1. One-dimensional models can be used for the vertical flow in a system of parallelhorizontal layers.

2. Two-dimensional models apply to a vertical plane while it is assumed that thegroundwater conditions repeat themselves in other parallel vertical planes. Spacing

equations of subsurface drains and the groundwater energy balance applied to drainage

equations are examples of two-dimensional groundwater models.

Figure 2.1 Two-Dimensional Modelling

3. Three-dimensional models like Modflow require discretization of the entire flowdomain. To that end the flow region must be subdivided into smaller elements (or

cells), in both horizontal and vertical sense. Within each cell the parameters are

maintained constant, but they may vary between the cells. Using numerical solutions of

groundwater flow equations, the flow of groundwater may be found as horizontal,

vertical and, more often, as intermediate.


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Figure 2.2 Three Dimensional Modelling

2.4 GROUNDWATER SOFTWARE OVERVIEW

Many different software tools are today available to help users to set up their models. The aim

of the models is to assist in the solution of practical problems, simulating processes in

subsurface fluids and porous media. In the majority of cases, modelling serves to improve

under-standing of hydrogeological systems. Forecasting and thus studying the response due to

different scenarios is the most ambitious goal of modelling efforts.

Software tools can be subdivided into different classes, for which codes that perform numerical

calculations are considered as the core software program. Around these packages have been

developed for several pre- and post-processing tasks. GMS, Visual MODFLOW, and PMWIN

are examples, which are built around the MODFLOW code in the core in most recent versions

accompanied by other numerical codes. Other packages, like FEFLOW, embed all tasks in one

package.

Some Softwares used for Groundwater Modelling are:-

Analytic Element Method FEFLOW SVFlux FEHM HydroGeoSphere MicroFEM MODFLOW

GMS Visual MODFLOW

OpenGeoSys


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SahysMod, Spatial agro-hydro-salinity-aquifer model, online: [8] US Geological Survey Water Resources Ground Water Software ZOOMQ3D

The most prominent code for groundwater modelling is MODFLOW. The most recent version

is MODFLOW2005, described by Harbaugh et al. (2000). The origins of MOD-FLOW can be

traced back to the beginning of the 1980s. An overview on the history of MODFLOW is given

by McDonald and Harbaugh (2003).

Figure 2.3 2-D view of a Groundwater Model


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3. MATHEMATICAL MODEL

3.1 GROUNDWATER FLOW

Groundwater flow models are based on the differential equations for groundwater flow.Such

differential equations, are usually based on Darcys Law as the linear macroscopicfl

uidmomentum balance equation, considering the drag terms of the Navier Stokes equation as

dominant, and on the principle of the fluid mass conservation.

In the year 1856, Henry Darcy, a French hydraulic engineer investigated the flow of water

through a vertical homogeneous sand filter. Based on his experiments, he concluded that the

rate flow through the porous media is proportional to the head loss and is inversely proportional

to the length of the flow path. .....Eq(3.1)

is the flow rate i.e volume of water flows through the sand filter per unit time.is the Hydraulic Gradient.is the coefficient of permeability.is the area of cross section.The Darcys Law was derived experimentally for one dimensional flow in a homogeneous

porous medium. The generalized three-dimensional form of the equation can be expressed as,

.....Eq (3.2)

Where,

,

, The negative sign also indicates that water is flowing from higher hydraulic head to lowerhydraulic head. Thus hydraulic gradient is negative along the direction of flow.Theequation

(3.2) can also be written as:

.....Eq (3.3)


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The flow in x, y and z direction can be written as:-

......Eq (3.4)

.....Eq (3.5)

.....Eq (3.6)

Depending on the special features of the situation to be modelled, different circumstances have

to be taken into account. A model for a confined aquifer is different from that for an unconfined

(phreatic) aquifer. The spatial dimensionality (1D or 2D or 3D) depends on the physical

situation and the aim of modelling. Depending on the very same aspects, a decision about

steady state versus unsteady simulations has to be taken, just to name the most basic properties

of a model.

There are different formulations of the differential equations. Equation (3.2) states the mass

balance in 3D:

.....Eq (3.2)

Where, denotes porosity, fluid density, v the three-dimensional vector of Darcy velocitythat is specific discharge, and represents mass sources or sinks of whatever type. Mostmodels work with a simplified version of equation (2), which is valid for constant density .

With the help of Darcys Law, the equation can be reformulated in terms of hydraulic head h.

Simplified 2D versions of equation (2) are used quite frequently, which are different for

confined or unconfined aquifers. In the confined situation:

.....Eq (3.3)

In whichandrepresent pumpingand rechargerates, respectively, where denotes thestorativityand the transmissivity.Usually the hydraulic head, h is the dependent primevariable, for which the differential flow equation is formulated and which is calculated by the

model.


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3.2 GOVERNING EQUATION FOR THE FLOW THROUGH PORUS

MEDIUM

3.2.1 CONFINED FLOW

Now, from the above two equations i.e. Darcys Law and Mass Flow equations all the equations

listed below are derived. So the Governing Equations for two dimensional flow through

confined aquifer is:- .....Eq (3.4)is the Aquifer thickness

his the piezometric

is the coefficient of Transmissibility (

)

is the specific storageis the Storage Coefficient ( [ As we are considering homogeneous aquifer condition]In generalised form we can write Equation 3.4 as:-

.....Eq (3.5)

Now for Steady-State Condition( Eq (3.5) becomes, 0

.....Eq (3.6)

This is Laplace's equation (2-D), the subject of much study in other fields of science. Many

powerful and elegant methods are available for its solution, especially in two dimensions.

Here,is known as Velocity Potential. The velocity potential may be defined as a scalar functionof time and space such that its derivative with respect to any direction gives the fluid velocity

I that direction.

.....Eq (3.7 a)


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.....Eq (3.7 b)

Now, lets consider 3-D flowin porous medium, so the main governing equation in 3-D flow

is (Non Homogeneous Aquifer):-

.....Eq (3.7)

Here,

Velocity Potential Considering Aquifer to be Homogeneous in nature so Eq (3.7) can be written as:-

Or

.

.....Eq (3.8)

3.2.2 UNCONFINED FLOW (Only 3-D Flow)

a. INHOMOGENEOUS ANISOTROPIC UNCONFINED AQUIFER 0.....Eq(3.9)

The right-hand side equals zero not because 0(it doesn't), but because in unconfinedaquifers Ss0.The flow domain for which solutions of this equation are sought is notconstant because the water-table position changes with time.


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b. HOMOGENEOUS ANISOTROPIC UNCONFINED AQUIFER

0Or. 0

.....Eq (3.10)

c.HOMOGENEOUS ISOTROPIC UNCONFINED AQUIFER 0.....Eq (3.11)

Note, this equation is again a Laplace Equation

3.3 GOVERNING EQUATION FOR RADIAL FLOW IN AN AQUIFER

The flow towards a well, situated in homogeneous and isotropic confined or unconfined aquifer

is radially symmetric. Fig. 3.1(a) shows the cone of depression caused due to constant pumping

through a single well situated at (0,0) in a confined aquifer. Fig.3.1 (b) shows the cone of

impression caused due to constant recharge through the well. In case of homogeneous and

isotropic medium, the cone of depression or cone of impression is radially symmetrical. The

governing equation derived earlier in Cartesian coordinate system for confined and unconfined

aquifer can also be derived for radial flow in an aquifer. In this lecture, we will derive the

governing flow equation for confined and unconfined aquifer in polar coordinate system. The

main objective of this conversion is to make the 2D flow problem a 1D flow problem. The

resulting 1D problem will be simpler to solve.

Figure 3.1 (a)Cone of depression (b)Cone of impression


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3.3.1 CONFINED FLOW

Let us consider a case of radial flow to a single well in a confined aquifer. Let there be the

radial flow towards a well and a control volume of thickness dr. The aquifer is homogeneous

and isotropic and have constant thickness of b. The hydraulic conductivity of the aquifer is K.

The pumping rate (Q) of the aquifer is constant and the well diameter is infinitesimally small.

The well is fully penetrated into the entire thickness of the confined aquifer.

So the flow equation for radial flow into a well for confined homogeneous and isotropic aquifer

is:- 1 .....Eq (3.12)

1 .....Eq (3.13)

In case of steady state condition, the governing equation becomes, 1 0.....Eq (3.14)

3.3.2 UNCONFINED FLOW (Only 3-D Flow)

Let us consider a case of radial flow to a single well. The unconfined aquifer is homogeneous

and isotropic. The hydraulic conductivity of the aquifer is K. The pumping rate (Q) of the

aquifer is constant and the well diameter is infinitesimally small. The well is fully penetrated

into the aquifer and hydraulic head in the aquifer prior to pumping is uniform throughout the

aquifer.


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Figure 3.2 an unconfined aquifer

So,the flow equation for radial flow into a well for unconfined homogeneous and isotropic

aquifer is:-

1 .....Eq (3.15)

Where,

Syis the specific yield which is equal to Ss/ h.

In case of steady state condition, the governing equation becomes,1 0.....Eq (3.16)


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4.0 NUMERICAL MODEL

There are different numerical techniques by which computer algorithms are derived from

equations that govern the model. In order to obtain a numerical model, the mathematical (e.g.

differential) formulation for continuous variables has to be transformed into discrete form. The

discrete variables (e.g. hydraulic head in flow models, concentration, or temperature intransport models) of the model are deter-mined at nodes in the model domain, determined by

a grid.

4.1 FINITE VOLUMES

The method of Finite Volumes (FV) is derived from a mass or volume balance for all blocks

of the model region. The load (e.g. volume or mass) balance in block i,j is obtained by:

..Eq (4.1)

where V denotes the volume or mass in the block, Qi, Qi+, Qj, Qj+ the fluxes across the

block edges, and Q other sources or sinks for volume or mass.

A set of equations for the values of the unknown variables in the block centers is obtained by

expressing the fluxes in terms of that variable. With the help of Darcys Law, all volume

balance equations come into a form that expresses relations between hydraulic heads fij= hij.

Similarly, Finite Volume grids may be of general form, as they are defi

ned by the budget offluxes across the boundaries of a block. The dependent variable is calculated at the center of

the block (block centered grid). Parameters are specified for blocks, that is, around the block

centers. The different form of grids used in FD and FV makes it difficult to compare results

obtained by the different methods without interpolation.

4.2 FINITE ELEMENTS

Finite elements grids can be of different shape, but very often they can be recognized by the

simple triangular form of the single elements. The triangular shape is convenient to

approximate arbitrary shaped regions with small deviations, where rectangular grids often

show stairway structures at least at parts of the boundaries.

Using Finite Elements, the solution of the differen-tial equation is found as a combination of

shape (or Lagrangian)-functions. These functions are different for dif-ferent elements. For the

most common triangular elements, the prescribed Lagrangian functions are linear within each

element. Thus f is approximated, for exam-ple, in Cartesian coordinates, by

, Within element..Eq (4.2)


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Figure 3.Aquifer showing A, well field and aquifer boundaries;

B, finite-difference grid; and

C, finite-element grid used in computer modelling of ground-water systems

4.3 FINITE DIFFERENCES

The method of Finite Differences (FD) is derived as approximation of the differential equation.

Derivatives (differential quotients) are replaced by difference quotients. For first and second

order derivatives, the simplest central stencils (CIS = central in space) are given by:

,

2 /

..Eq (4.3)


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where the f -values denote function values at the grid nodes, that is, fi is the approximate value

of the function at node i, at the previous node, andat the following node (see Figure1). This leads to a system of equations for the unknown values (fi ,i = 1..N),where N denotes

the total number of nodes. For transport problems, the upwind scheme (BIS = backward in

space) is important:

..Eq (4.4)

4.1 SOLUTION OF STEADY STATE FLOW EQUATION FOR CONFINED

HOMOGENEOUS AND ISOTROPIC AQUIFER (FOR FLOW THROUGH POROUS

MEDIUM)

The steady state flow equation for homogeneous and isotropic confined aquifer can be written

as :-( If a sourceN(x, y, t)is present, the equation becomes)

, 0

..Eq (4.5)

Here, the source term represents the pumping or recharge per unit horizontal area of the aquifer.

If case of pumping a negative sign is used as mass is withdrawn from the control volume. On

the other hand, the source term will be positive in case of recharge as we are adding mass to

the system. For example, if Qm3/sec is the pumping rate from the control volume, the value of

N(x, y, t) will be .

Where, T is the transmissivity of the aquifer (m2/day), is the hydraulic head (m), N is the

pumping or recharge value (m3

/day/m2

).


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Figure 4.2 2D confined aquifer with boundary conditions

Now consider the aquifer as shown in Fig. 4.1. The 2D confined aquifer is homogeneous and

isotropic and has no flow boundary at two sides and constant head boundary on two other two

sides as shown in the Fig.4.1. For applying the finite difference scheme, the aquifer has to

discretize as sown in Fig. 4.2 below. Let and are the size of a discretize grid.

Figure 4.3Discretized 2D confined aquifer with boundary conditions


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,

2,

,

,

2,

,

..Eq (4.6)

Putting (16) in equation (15) and after simplifying, the finite difference approximation of the

steady 2-D flow equation at cell (i, j) may be expressed as:

, , , , 2 2, , 0..Eq (4.7)

Where,

and

4.3 SOLUTION OF STEADY STATE FLOW EQUATION FOR CONFINED

HOMOGENEOUS AND ISOTROPIC AQUIFER (FOR RADIAL FLOW)

4.3.1 CONFINED AQUIFER

In case of steady flow in confined aquifer, the flow equation becomes:-

1 0

Figure 4.4: Confined aquifer


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Now, by using Darcys Law our solution will be:-

2

..Eq (4.8)Knowing hydraulic head at the well, the equation (4.6) can be used to calculate steady state

hydraulic head for any values of .

4.3.2 UNCONFINED AQUIFER

In case of steady flow in unconfined aquifer, the flow equation becomes,

Figure 4.5: Confined aquifer and boundary condition

By, solving equations we have,

log

..Eq (4.9)

Knowing hydraulic head at the well, the equation (4.7) can be used to calculate steady hydraulic

head for any values of r. This equation can also be used for estimation of aquifer conductivity.

The equation can be written for calculating aquifer conductivity as,

log


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4.4 CONCEPT OF FLOW NET

A flow net is a graphical representation of two-dimensional steady-state groundwater flow

through aquifers. Construction of a flow net is often used for solving groundwater flow

problems where the geometry makes analytical solutions impractical.

The flow lines and the equipotential lines are drawn by trial and error. It must be remembered

that the flow lines intersect the equipotential lines at right angles. The flow and equipotential

lines are usually drawn in such a way that the flow elements are approximately squares.

Drawing a flow net is time consuming the tedious because of the trial-and-error process

involved. Once a satisfactory flow net has been drawn, it can be traced out.

Figure4.6Flow net around a single row of sheet pile

Calculation of seepagefrom a flow net under a hydraulic structure. A flow channel is the strip

located between two adjacent flow lines. To calculate the seepage under a hydraulic structure,

consider a flow channel as shown in Figure 4.6. The equipotential lines crossing the flow

channel are also shown, along with their corresponding hydraulic heads. The portion between

the two successive flow lines is known asFlow Channel and the portion enclosed between the

two successive equipotential line and successive flow lines is known as field.


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Let b and lbe the breadth and length of field.

head drop through the field. discharge passing through the flow channel

H = total hydraulic head causing flow.

Then from Darcys Law of flowing through the soils:

1..Eq (4.10)

IF,

total number of potential drops in the complete flow net Hence,

..Eq (4.11)

Mathematically, the process of constructing a flow net consists of contouring the two harmonic

or analytic functions of potential and stream function. These functions both satisfy the Laplace

equation and the contour lines represent lines of constant head (equipotential) and lines tangent

to flowpaths (streamlines). Together, the potential function and the stream function form the

complex potential, where the potential is the real part, and the stream function is the imaginary

part.

In the previous section we have already discussed about the concept of Velocity Potential, now

solution of Laplace equation gives two sets of curves known as equipotential lines and

stream lines, mutually orthogonal to each other. Stream Function is defined as scalarfunction of space and time such that a partial derivative of this function with respect anydirection gives the velocity component in a direction +90 to the original directions.

,

..Eq (4.12)


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Now, we know that,

,

So, according to Cauchy-Riemann equation,

..Eq (4.13 a)

..Eq (4.12 b)

In solving groundwater problems, we need to determine only one of the two functions

, the other function will then follow from the relationships of equations 4.12.

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