25 CHAPTER 3 − MATHEMATICAL MODEL A mathematical model is a set of equations that describe and represent the real system. This set of equations uncovers the various aspects of the problem, identifies the functional relationships between all the system’s components. These equations could be algebraic, differential, or others, depending on the nature of the system being modelled. In general, to obtain a way to control or manage a physical system, a mathematical model is introduced and solved incorporating the boundary conditions and the solution is used to determine the behaviour of the physical system. Conceptually, the modeling techniques used for system representation can be very simply explained as below and schematically as in Figure 3.1 Fig 3.1 Schematic representation of conceptual modelling technique • Select or formulate a suitable model • Assume the parameters approximately • Adopt some error function to quantify the difference between measured and predicted responses • Minimize the error function • Determine the parameters accurately • Predict system response 3.1 Systems and Modelling Concepts It is impossible to carry out experiments and tests on real water resources system in order to determine their behaviour under different conditions. It is necessary to consider models of the system. We manipulate the model and use the results obtained from the model for making decisions regarding the real system. The model chosen to represent the real system should be representative and simple. Yet it should not be too simple to make the solution not applicable in reality. If we make the models too complex we may not be able to solve the mathematical equations involved. Further, it is meaningless to choose such a model, which can give very accurate results where the input data is much less accurate. Actual System Response Model predicted System Response Mathematical Model Modeled Input Non - Modeled Input Real Physical System Solution Strategy (Optimization)
Transcript
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CHAPTER 3 − MATHEMATICAL MODEL
A mathematical model is a set of equations that describe and represent the real system. This set
of equations uncovers the various aspects of the problem, identifies the functional relationships
between all the system’s components. These equations could be algebraic, differential, or others,
depending on the nature of the system being modelled.
In general, to obtain a way to control or manage a physical system, a mathematical model is
introduced and solved incorporating the boundary conditions and the solution is used to determine the
behaviour of the physical system. Conceptually, the modeling techniques used for system
representation can be very simply explained as below and schematically as in Figure 3.1
Fig 3.1 Schematic representation of conceptual modelling technique
• Select or formulate a suitable model
• Assume the parameters approximately
• Adopt some error function to quantify the difference between measured and predicted
responses
• Minimize the error function
• Determine the parameters accurately
• Predict system response
3.1 Systems and Modelling Concepts
It is impossible to carry out experiments and tests on real water resources system in order to
determine their behaviour under different conditions. It is necessary to consider models of the system.
We manipulate the model and use the results obtained from the model for making decisions regarding
the real system. The model chosen to represent the real system should be representative and simple.
Yet it should not be too simple to make the solution not applicable in reality. If we make the models too
complex we may not be able to solve the mathematical equations involved. Further, it is meaningless
to choose such a model, which can give very accurate results where the input data is much less
accurate.
Actual System Response
Model predicted System Response
Mathematical Model
Modeled Input Non - Modeled
Input
Real Physical System
Solution Strategy (Optimization)
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There is no unique model for a given region. One should have a hierarchy of models of
increased refinement and the appropriate one from the point of view of need, cost and availability of
data, should be selected in each case.
The selected model must be calibrated to determine its parameters. If such information is
already available, these can be refined at calibration stage by trial and error or optimization.
3.1.1 Numerical models
Numerical models using digital computers are currently the major modelling techniques used for
groundwater system studies. Much effort has been made in many parts of the world in the
development of techniques for numerical solution of the partial differential equations that govern the
flow of water in aquifers. Many computer programs have been developed and published (Prickett,
1971; Lonnquist, 1971; Thomas, 1973 and Trescott et al., 1975).
The two numerical methods that are commonly used are the finite difference methods and the
finite element methods.
3.1.2 Finite difference method (FDM)
The fundamental equation governing two dimensional flow of water in saturated non
homogeneous anisotropic aquifers is given as
Where Tx, Ty are the aquifer transmissibility in x and y direction, h is the head, S is the storitivity
of the aquifer, and Q is the net groundwater withdrawal rate per unit area of the aquifer.
The finite difference equations can be derived either by replacing the derivatives in the above
differential equation by their finite difference approximation or from physical standpoint involving the
conservation of mass and Darcy’s law (Chawla, 1990).
3.1.3 Integrated finite difference method (IFDM)
A modified form of FDM is called as IFDM. This method also can be used for groundwater
system models. Unlike FDM where square or rectangular grids are used, this utilizes an arbitrary grid.
In IFDM the region of interest is divided into smaller polygonal areas, where each has a node
which is used to connect each area mathematically with its neighbours. It is assumed that all recharge
and abstraction in a nodal area occurs at the nodes. In other words, each node is considered to be the
representative of its nodal area. For each node a certain storage coefficient or specific yield value is
assigned, which is constant and representative for that nodal area. A certain hydraulic conductivity is
assigned to the boundaries between nodal areas, thus allowing directional anisotropic condition.
3.1.4 Polygonal network
This method assumes a linear variation of the measured values between each pair of adjacent
observation points. Perpendicular bisectors of the lines connecting adjacent observation points form
the polygon corresponding to that point.
QthS
yhTy
uxhTx
x±
∂∂
=
∂∂
∂∂
+
∂∂
∂∂
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The model is based on the two well known equations, Darcy's law and the equation of
conservation of mass. The equation of continuity of an unconfined aquifer in which there is no vertical
variation of properties will be (Bear, 1977)
Applying Darcy's Law
Hence the above equation can be rewritten as
Where k is the permeability between polygonal boundaries and transmissibility T is equal to
k*m.
Analytical solution of the equation is not possible for a complex natural system. Hence,
numerical methods are necessary to solve the said equation. The finite difference form of this
differential equation for a typical node B and its association with its neighborhood node i as given in
Figure 3.2 can be reduced by implicit numerical integration technique corresponding to the time
interval j and j+1 with time step ∆t to the equation shown below (Chun et al., 1963):
Observation well
Fig 3.2 Typical polygon for node B
0)()( =+++ QSymVymVx dtdh
dyd
dxd
dxdhkVx −=
dydhkVy −=
0)()( =++−− QSykmkm dtdh
dydh
dyd
dxdh
dxd
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Where, hi - Piezometric head of node i
hB - Peizometric head at node B
YiB = (JiB/LiB) - Conductance factor
TiB - Transmissibility at mid point between node B and i
JiB - Length of perpendicular bisector associated with node B and i.
LiB - Distance between nodes i and B
AB - Polygonal area of node B
SB - Storage coefficient of node B
QB - Volumetric flow rate per unit area at node B.
M - No of observation wells surrounding node B
∆t - Time step between j and j+1
If the assumed values of transmissibility, storage coefficients, recharge coefficient for irrigation
scheme (aB), recharge coefficient for irrigation field (bB), recharge coefficient for rainfall (cB), and the
withdrawal factor for agro and domestic pumping (dB) are correct, subsurface flow + vertical flow will
be equal to change in storage. But due to the error in assumptions there will be residue RESBj+1.
Where
3.2 Models for Management and Conjunctive Use
Physical, social, legal and economic factors determine the operation of conjunctive ground –
surface – rain water systems. The relative importance of the interacting parts of the total system
1111
1
1 )()( ++∆
++
=
+ +−−−= ∑ jBBB
jB
jBt
SiBiB
jB
ji
M
i
jB QAAhhTYhhRES B
.)()( 1111
1
++∆
++
=
+−=−∑ jBBB
jB
jBt
SiBiB
jB
ji
M
iQAAhhTYhh B
iBiBj
Bj
i
M
i
jB TYhhQQflowSubsurface )(_ 11
1
1 ++
=
+ −== ∑
)(.__ 11 jB
jB
BBj hhtSA
BSTORstorageinChange −∆
== ++
11111_ +++++ −++== jdB
jirB
jifB
jisB
jBB QdQcQbQaQAflowVertical BBBB
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causes different levels of complexity in different systems. Of the many interacting parts of a system,
the physical characteristics are often relatively well understood. The same applies to economic and
legal aspects too.
In most analyses, the legal characteristics of the system have been applied as a set of
constraints. A major difficulty arises when transferring laws and regulations overwhelm the other
characteristics of the system and dictate the policy for conjunctive use operation. However, by
studying the system sensitivity to legal constraints, their impact on the overall operation and their cost
can be determined. Some of the types of models used in literature for system management and
conjunctive use operation are described here.
3.2.1 Optimisation models
These models have been very popular among water resources planners. Whereas a simulation
model seeks to reproduce the dynamics of the system, an optimization model seeks to obtain the best
fit for the observations. Linear programming, dynamic programming, and nonlinear programming are
the main tools of optimization, which are discussed in brief below.
3.2.1.1 Linear programming
Linear programming is a method of system optimization in which all the operations can be
approximated by linear equations. The two parts of a linear optimization model are the objective
function and the constraints (Richard, 1989).
This type of problem is solved with the help of simplex algorithm. The simplex algorithm for a
straight forward linear optimization model is carried out as follows.
Step 1: Convert the objective functions to maximize and all the constraints to less than
inequalities.
Step 2: Add slack variables to all the constraints. This, in effect, makes all the variables zero, and
places one at the origin for initial external point (initial feasible solution). These slacks are called
basic.
Step 3: Check the objective function to see if there is a nonbasic (zero value) variable that would
increase the value of the objective function. If so, it should be brought into the basic; if not, one
has reached the optimal solution and so can stop.
Step 4: Calculate how large the new basic variable from step 3 is made without becoming
infeasible and which of the old basic variables it will replace in the basic. This variable and the
respective constraint comprise the pivot element.
Step 5: Pivot; i.e. by algebraic manipulation (like Gaussian Elimination), drive to zero all pivot
elements in the column of the new basic variable and make the pivot element 1.0 by dividing that
constraint by the original pivot element (coefficient).
Step 6: Go back to step 3.
3.2.1.2 Dynamic programming
Dynamic programming (DP) is specially suited to the class of problems which require sequences
of optimal decisions. The sequences may be over time or over space. In DP, one usually starts from
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the end and work back to the beginning of a procedure called recurrasion. Each sequence is called a
stage, and the situation at each stage is called the state.
Dynamic programming has the advantage of allowing almost any kind of objective function. The
objective function may be non-linear or even discontinuous. However, there are severe restrictions on
the number of decision variables.
3.2.1.3 Non linear programming
Classical nonlinear programming can be divided into constrained or unconstrained models. For
each division, the objective function may be convex, concave or nonconvex. When the objective
function is nonconvex, one may not be able to guarantee the solution.
If the objective functions in nonconvex, this may only be a local stationary point. Consequently, it
is helpful to determine the convexity of nonlinear objective functions. If the entire odds are
nonnegative, the objective function is convex; if the odd ones are nonnegative and even ones are
negative, the function is concave. If neither of these conditions holds, the function is nonconvex
(saddle point). For constrained nonlinear optimization, the Lagrangian equation is the classical
approach.
3.3 Aquifer Simulation Model and Inverse Modelling Technique
The aquifer simulation model developed by using the integrated finite difference method can be
used to determine the response of the aquifer to various management policies such as operational
policy of minor and medium irrigation schemes location of pumping wells, recharge wells, the quantity
to be recharged, etc. Before application the model should be calibrated using observed data (Chun et
al., 1963).
During calibration the aquifer parameters such as Transmissibility, Storitivity, Recharge
coefficient etc, are so adjusted that the computed levels match with the observed levels.
If there are results of some pumping tests in the aquifer they should be used. This process of
determination of aquifer parameters is also called inverse modelling (Chawla, 1990).
3.3.1 Determination of aquifer parameters.
The basic idea of inverse modelling consists of using past information on both aquifer stresses
such as pumping and aquifer responses such as water levels and determining the values of the
aquifer parameters which will cause the model equations relating the two sets of data to be satisfied.
The net withdrawal rate N is equal to L+R-P where L is the natural recharge R is the artificial recharge
and P is the pumping. In the forecasting problem, we know the aquifer parameters Tx(i,j), Ty(i,j), S(i,j)
and natural replenishment L(i,j,k) for a period of time in the past and we seek a solution of the above
equation for the aquifer parameters Tx(i,j), Ty(i,j) and S(i,j).
3.3.2 Regional aquifer parameters
Pumping tests determine aquifer parameters in the vicinity of pumping wells. Regional values of
aquifer parameters may be different from the local values obtained by pumping tests. Regional values
are better determined by calibration of the model (Yoganarasimhan, 1979). In this method a trial set of
aquifer parameters is used in the model. The calculated response of the aquifer model then is
compared with the observed value in the field and, if the computed values do not match with observed
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values, the aquifer parameters are suitably modified. This process is repeated until the calculated and
observed value match. The main disadvantage of this trial and error method is that it does not
incorporate any algorithm for approaching the best solution in a systematic way (Bear, 1977).
Attempts have been made to obtain optimal set of aquifer parameters by minimizing one or
more error criteria. In the following section derivation of an optimal set of aquifer parameters by using
a single error criterion is described.
3.3.3 Optimal set of aquifer parameters
It is desirable to discrete the given region into a number of aquifer cells or polygonal areas for
the purpose of determination of aquifer parameters. Known geologic information should be used in
deciding the number and size of cells or polygons.
Different criteria for error minimization can be used. Following are the commonly used criteria.
• Minimize the sum of the absolute values of all deviations
• Minimize the sum of the square of the deviations
The first one leads to a linear programming problem. And the second one leads to a quadratic
programming problem.
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CHAPTER 4 − MODEL FORMULATION
4.1 Description of the Model and Assumptions
The model used in this research is an aquifer simulation model developed by using the
advanced numerical modelling technique called Integrated Finite Difference Method (IFDM). The
model is calibrated by the technique called optimal set of aquifer parameters as explained in chapter 3
The following assumptions are incorporated in this model for the restricted catchment selected
for this study.
4.1.1 Hydraulic assumptions
• The aquifer is treated as a two-dimensional flow system.
• Only one aquifer system is modelled with one storage coefficient in the vertical direction.
• The aquifer is bounded at the bottom by an impermeable layer.
• The upper boundary of the aquifer is an impermeable layer (confined aquifer) or a slightly
permeable layer (semi confined aquifer) or the free water table (unconfined aquifer)
• Darcy’s law (Linear resistance to laminar flow) and Dupuit’s assumption (vertical flow can be
neglected) are applicable for the aquifer under study.
• The processes of the infiltration and percolation of rain and surface water and of capillary rise
and evapotranspiration, taking place in the unsaturated zone of the aquifer (above the water
table) need not be simulated. This means the net recharge to the aquifer must be calculated
manually and fed to the model (Chawla, 1990)
• The recharging period of eight months was taken as from 1st October to 31st May of the
following year and discharging period of four months was taken as from 1st June to 30th
September (Shanmuhananthan, 2004)
• The minor discharge in February and March need not be simulated (Chawla, 1990)
The last two assumptions are also justified by the monthly water level collected from April 2004 to
December 2004 and from January 2005 to December 2005 are given in table 4.1 and table 4.2. The
monthly water level fluctuations of few observation wells are given as figure 4.2 and 4.3.
4.1.2 Operational assumptions
• Same groundwater elevation within a polygonal area.
• The area where the minor & medium Irrigation schemes are governing the water table, then
one meter below FSL of the tank can be taken as the water table elevation
• The rain fall of Vavuniya can be used for the entire area under study as all the polygons are
around Vavuniya rainfall station and within 16 km radius.
• The Irrigation efficiency can be assumed as 70 %( Ponrajah, 1984).
• Conveyance efficiency of the canal can be taken as 80% (Ponrajah, 1984).
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• As the Irrigation canals within this study area are very small and for simplicity, recharge from
canal and irrigation field can be combined for calculation
• All the 6 medium and 41 minor Irrigation schemes within the study area have their maximum
head of water less than 3 m. Hence the percolation of water is calculated either as 0.005
m/day/plan area (Rushton, 1990) or 0.5% of the volume stored monthly (Fernando and
Shaktivadival, 1994)
• Cropping intensity can be taken as 1.25 (Department of Irrigation, Administration Report,
1999 to 2002)
• To keep 10% of the full capacity of the minor and medium irrigation schemes at the end of the
cultivation, 12% of the cultivation has to be forgone. (An electronic version of a program
prepared for irrigation scheme operation study to justify this assumption is annexed)
4.1.3 Economic assumptions
• Benefit cost ratio based on present value should exceed unity (Ponrajah, 1984; Planning
Commission of India, 1961 and Irrigation Commission of India, 1992)
• Percaptia domestic water consumption can be taken as 160 litres/day (NWS&DB)
• Water requirement for OFC can be taken as (DOA administration report , 1999 to 2002)
o Maha 0.457 to 0.61 ha.m/ha.
o Yala 0.61 to 0.76 ha.m/ha.
• Net return from one hectare of paddy cultivation will be Rs. 10300/= (Annexure1)
• One meter raise in water table will save 1.4 unit of electricity for the pumping of 10 m3 of water
(Annexure 2)
• One kilometre of peripheral treatment will cost Rs. 3.3 million in 2007 rate. (Annexure 3)
4.2 Study Area
The rationale behind the selection of this study area (as in Figure.4.1) is that the aquifer of this
region very well reflects the typical groundwater problems of unconfined aquifers in shallow weathered
and rarely fractured rock with thin soil mantle areas. Study of this nature of problems is the prime
intention of this research. This area falls within the boundary of Vavuniya and Vavuniya South
Divisional Secretary’s divisions. The Grama Nilathari Divisions coming within the study area are given
below in Table 4.3.
The intention in the selection of this area for this research is to examine the behavior of
groundwater flow in a restricted aquifer and to come out with an appropriate strategy to ensure
sustainability in groundwater management policy for any restricted region by changing the operational
policy of minor/medium irrigation schemes within the region.
In addition to the above, the availability and accessibility of reliable sources of physical and
hydro geological data required for the research is also one of the prime reasons for the selection of
this area.
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4.2.1 Location and size
The study area is located in the northern part of Sri Lanka between 9o 22' and 9o 52' North
latitude and between 79o 52' and 80o 49' East longitude. The area covers 6 major tanks, 40 minor
tanks (Table 4.4), around 2000 shallow wells including 41 observation wells (Table 4.5) and covering
185.23 km2 in both Vavuniya and Vavuniya south Divisional Secretary’s Divisions in Vavuniya District.
1 Eng S S Sivakumar’s house. 38/39 D Pilliyar Kovil lane Ukkulankulam Vavuniya 2 Nelukulam common well 3 Patthiniyar Makilankulam- Jeyasinganathan house 4 Marakalai house - furniture shop owners house 5 Vavuniya - Irrigation quarters 6 K.Sithamparam - opposite of Skanthapuram school 7 Koomankulam - Nagentheran house
Turn 4th mile post of Mannar road to sooduventhapulavu road through Arapunagar road
19 Anpu illam, Mannar road 5th mile post 20 Mannar road 4th mile post, opposite the lane of G.S house, Jack tree house 21 Thurairajah house, Kaneshapuram 22 Puthukkulam F.O building 23 K.Amuthalingam house, Shanthasollai housing project 24 M.Vimalathasa, Mamadu road 4th kilometer, Sarvothayam building 25 Santhanakunasegara, Mamadu Nelukulam 2nd mile post 26 Maduganthai hospital of arulvetham 27 Thetkiluppai kulam kovil 28 Ellapparmaruthan kulam - Agricultural school 29 Iratperiyakulam - pansallai 30 Sooduventhapura mosque 31 K.Uruthirasingam, 7th mile post Mannar road 32 R.D.F building (U.N.H.C.R) Pampamadu 33 House near the Suntharapuram school 34 Shasthirikoolan kulam kovil 35 Maharampaikulam housing project common well next to school 36 7 th kilometer kovisana seva building Akkpopura Mamadu 37 Ariyaratna house next to sub post office 38 Madukanda common well in playground 39 The House next to 7th km post 40 Catholic church - Periya Koomarasankulam 41 Arugandilvelli , Iratperiyakulam – Pooduventhapura road
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4.2.2 Climate
This area falls within the dry zone of Sri Lanka and in the Agro-ecological region of DLI
(Ponrajah, 1984). Average annual rainfall of the district is around 1400 mm. The monthly average
temperature is around 27.5o C and it is found lower than this during October to January. The main
rainy season extends from early October to late January and the sub rainy season extends from late
March to late May. Hence the recharging period of eight months is from 1st October to 31st May of the
following year and discharging period of four months is from 1st June to 30th September, ignoring the
minor discharge in February and March (Chawla, 1990). The graphical water level fluctuation pattern
is given below in figure 4.2 and 4.3.
Fig. 4.2 Monthly Groundwater Level Fluctuation
85.0087.0089.0091.0093.0095.0097.0099.00
101.00103.00
Apr-04
Jun-0
4
Aug-0
4
Oct-04
Dec-04
Feb-0
5
Apr-05
Jun-0
5
Aug-0
5
Oct-05
Dec-05
Wel
l wat
er le
vel (
m)
Node 4Node 5Node 6Node 7Node 8Node 9Node 10
Fig, 4.3 Monthly Groundwater Level Fluctuation
66.0068.0070.0072.0074.0076.0078.0080.0082.00
Apr-04
Jun-0
4
Aug-0
4
Oct-04
Dec-04
Feb-0
5
Apr-05
Jun-0
5
Aug-0
5
Oct-05
Dec-05
Wel
l wat
er le
vel (
m)
Node 41Node 35Node 30
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The average rainfall for the recharging and discharging periods for the study period are given in
Table 4.6.
Table 4.6 Average seasonal rainfall
Period of months considered as seasons for this study Season No Seasonal rain fall
