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Good morning, and welcome to this the lecture number 37 of the course Water Resource Systems
Modeling Techniques and Analysis. Over the last few lectures, we have discussed the fuzzy
optimization technique, which is especially useful, when there is a conflict in the water
resource systems, which typically exists, because there are several stakeholders in
any water resource systems problem.
Specifically, what we dealt with in the previous lecture that is the last lecture was to deal
with a situation, where we have a storage target and a demand, and the release target;
and these two are conflicting in general, because if you make more release, then the
storage will be depleted. So if you want to maintain both the storage target as well as
the release target, one way of doing that was through our deterministic crisp optimization,
where you will take the minimized square deficit like this, and then panelize the T S itself;
here T S is the storage target, and you were maximizing the storage target. And that is
how you get you obtain the maximized storage target, and the associated storage is during
each other time period for given inflow sequence; and this also decides on the releases for
a given K, that is the storage capacity. And then you are looking at the deviations of
the release from the target, that is a only only whenever there is a deficit release from
a target. So this was your crisp problem. This you converted into a fuzzy optimization
problem by defining fuzzy membership function for the storage as well as for the release.
Now for the storage, what did we do? We said as it approaches, as the storage S t during
time period t, approach is the storage target T S, which is again a decision variable, then
the membership function becomes close up to 1; as it deviates away from T S, the membership
function will be close up to 0 on either side of T S. Whereas on the release, what we said
is that if the release is greater than the T R, our membership function is 1. Now we
constructed these membership functions, and then looked at maximize… Looked at the objective
function, where we are maximizing the minimum of these membership functions mu S t and mu
R t. So, the decisions are S t and R t, and the associated membership functions are mu
S t and mu R t; we are looking at the set of S t and R t, which satisfies all these
conditions which will in fact, maximize the minimum value associated with those particular
membership functions mu S t mu R t. So, this is the way we formulated the fuzzy
optimization problem. We looked also at the comparison of solutions that we obtained from
fuzzy optimization problem with crisp optimization problem for another example problem. So with
that now, we more or less complete the techniques that we have to learn for water resources
systems analysis. So, right from beginning if you recall, we have been focusing on the
techniques of problem solutions; for example, we started with constrained and unconstrained
optimization using the methods of calculus. Then we went on to the linear programming,
and then the dynamic programming, simulation recover, and then after that we went into
probabilistic optimization problems or the stochastic optimization problems, where we
dealt with chance constrained linear programming as well as stochastic dynamic programming.
Then towards the end, I have introduced the fuzzy optimization problem.
Now, these are essentially the techniques or the tools with which we address a problems,
and as we have gone along, we are also dealt with reservoir operation problems, multi reservoir
operation problems, and water quality problem in the fuzzy optimization and so on so. So
some of the applications we have dealt with, but the main focus was all through on the
techniques themselves. So, from this lecture onwards what we will
do is, over the next four lectures including today’s, we will take up some specific model
formulations for specific problems dealing with water resources systems. In the process,
I will also discuss some case studies, but the emphasis again is on model formulations
using the techniques that we have gone through in this particular course.
So, in today’s lecture, we will introduce the important model formulation for conjunctive
use of surface and ground water. Now this is we are talking about specifically large
river basis or large water resource systems, where there is a source of surface water available,
there is a source of ground water available, you need to manage both these systems. So,
ground water system as well as surface water system together or conjunctively, such that
you are able meet the demands, that are necessary; yet at the same time, the system is sustainable;
in the sense that, the ground waters do not depict beyond certain level; at the same time,
the ground water does not enter into the root zone or the it does not come nearer to the
ground surface causing salinity problem, water logging problems etcetera. And in the deficit
situation of surface water, you are able to use the ground water to meet the demands.
So, ground water and surface water together, you need to plan; you utilization of ground
water as well as surface water over a period of time. Now, this is the extremely relevant
in countries like ours, where there are large number of religions which receive deficit
rainfall in many years. And then we need to use the ground water resources during the
deficit rainfall period; and yet at the same time rich as the ground water wherever there
is a whenever there is a excess rainfall. So, these two together that is ground water
system as well as surface water system, we need to look at a joined or conjunctive use
of both these resources together. Now, the type of model that I will introduce
is extremely simple, and the first cut type of model just to understand how we formulate
the models or such problems. It can in fact, be as complex as the details that you would
like to input into the model, which all presently discuss, but because this is a first time,
I am talking about conjunctive use; let us look at some physical features of the conjunctive
problems, and then see what are the advantages, what are types of constraints that we need
to meet and so on.
So, what does the conjunctive use imply? It implies first of all, we impound the stream
water in a surface reservoir, and then we transfer this surface water at some optimum
rate to the ground water reservoir. So, we are meeting the demand, yet at the same time,
we are also recharging the ground water. Then whenever there is a above normal precipitation,
we use the surface water to the extent possible to meet the demands; and then the excess water
we use to recharge the ground water; we may have artificial recharge structure, through
which we recharge the ground water. And then during the drought periods, where there is
deficit rainfall, you have a large amount of ground water storage available, which has
been recharged during the excess rainfall period; and we start using the ground water,
during the deficit period. So, ground water actually is a sort of insurance
against the droughts; and therefore, we need to look at the ground water as a result storage,
in which we keep on putting additional water that is available from the surface sources,
whenever there is a excess rainfall or excess surface water available. Now, during the drought
periods what happens? Because you do not have surface water sources, you start using more
of ground water resources and therefore, the ground water starts depleting.
Now, when we are operating the ground water system, in conjunction with the surface water
system, we need to make sure that the ground water levels will fluctuate within an acceptable
band; that means, they should not cross an upper limit, because that will cause water
logging; they should not cross a lower limit that becomes unsustainable, because the ground
water levels will be lower to extends beyond possible usage.
Then you must also realize here that the surface water reservoirs once they are silted up,
typically in the period of about 100 years or 150 years in some cases, the surfaces water
reservoir gets silted up. Then you need to look for alternate surface water sources;
whereas the ground water storage by enlarge is the stable; it is not dependent on how
the development is taking place on the surface of the ground; the ground water storage is
more or less fix for a particular region. So, yield from ground water sources is there
for more dependable than that from the surface reservoirs; if you are able to manage the
ground water resources well, you can depend more on the, that is the reliability of ground
water supply can be increased. Also the physical and chemical quality of ground water is more
uniform than that of surface water.
Now these are some of the advantages; however, in the coastal regions typically, if there
is saline ground water that is salinity injects takes place and therefore, the ground water
become saline, in which case you need to inject more of fresh water or more of surface water,
so that the salinity gets reduced. Now of course, it is not practically possible to
divert all of surface water to underground, even if the operation were profitable; what
I mean by that is that you have a surface water source, and then you may have certain
means or mechanism by which you want to put the fresh water into the ground water, but
all of the surface water cannot obviously, be put into ground water, because of several
constraints.
Now, you look at a irrigated area typically, when we are talking about conjunctive use
of surface and ground water especially, in our country we talk about irrigation, use
of water for agriculture, which is the irrigated agriculture. But this needs to be looked at
in a slightly broader context although; now with water being used more in the urban areas
especially, the ground water being used more in the urban areas, it is important for us
to look at conjunctive use of surface as well as ground water sources in urban context also.
But as far as this course is concerned, this lecture is concerned, I will focus mostly
on the use of ground as well as surface water for irrigated agriculture.
So typically, we have an irrigated area, you may have the bore wells or tube wells from
which you are using the ground water; in addition you may be supplying the water through surface
source for example, this is the canal, and from the canal, you have branch canals and
field canals etcetera. So, you have a canal network also supplying water from the surface
sources. You need to decide, how many such wells are necessary in this irrigated area,
and what is the level of pumping that you need to do from each of these wells? Across
each time period within a year such that along with the surface water source, you are able
to meet the demands to the best extent possible. So, that is the decision that you need to
make. So, the question is how much of surface water,
and how much of ground water we need to apply, such that this system becomes sustainable
in the longer. Now what do you mean by sustainability etcetera, we will slightly defer the discussion;
but essentially, it means that we are able to maintain this level of irrigation by using
surface water and ground water optimally over a long period of time. So, that is the idea.
So, for a long period of time, we should able to meet these demands satisfactory.
Now, during the excess period I just mentioned that you may divert the excess rainfall or
excess surface water into detention ponds; so you may create some artificial detention
ponds and divert this water, so that the water seeks through, and then adds through the water
table. In the conjunctive use, we must also be alert to the undesirable effect, undesirable
effects of excess use of ground water or lowered use of ground water, which will which will
bring the ground water to near surface. So, there are two issues that you need to
keep in mind especially, when you are dealing with large irrigated areas. If you do not
at all use the ground water, and simply you are using the surface water, from the surface
water irrigation, there is a recharge that is taking place to the ground water, as the
water flows through the canals, there is seepage, and part of the seepage and in fact, most
of the seepage may contribute to the ground water. As the result of which, the ground
water levels will be continuously increasing, if you do not use the ground water at all.
And then the ground water may enter into the root zone of the crops, because we are talking
about the agriculture irrigated that is the irrigated agriculture. Once a ground water
enters a root zone, it causes salinity problems, water logging problems, which affects the
irrigation; and therefore, you need to manage the ground water as well as the surface water
systems, such that the ill effects are nullified or ill effects are taken care of.
So, salivation is one of the effects that you need to be alert; then increased operation
cause, what do we mean by this, is that if we were only putting the surface water systems,
then there is a initial capital cost for creating the infrastructure. But the operational cost
would be minimal, because the water flow by gravity, and once the infrastructure is in
place, you do not have to do much about the surface water resistance; whereas if you want
to develop the ground water, you need to have energy cost, associated with the ground water
usage. So, there is a increased operational cost, if you start using ground water. Then
there is a danger of overdraft of ground water basics. So, we need to look at all of these,
when we are looking at the conjunctive use of surface water and ground water.
Now, for the systems models, when we are looking at the conjunctive use of surface and ground
waters, we may have several objectives; for example, the first level of objective that
we may want to fabulate is to meet the water demands at lowest overall cost. So, you want
to create a surface water infrastructure, and you want to create a ground water infrastructure,
to meet a certain demand pattern; and this infrastructure that we create along with the
operational cost must be minimum; yet at the same time, you should be able to meet the
demands; that is the first level of objective. Then you may have several other objectives
associated with the conjunctive use problem; for example, you may have a water quality
objective, where you may want to maintain the concentration levels of water quality
constituents, both in the surface water as well as the ground water. So, you may have
quality objectives, which we can be made through, which can be studied through the system’s
models. Then as I just mentioned, there may be undesirable effects; over draft of ground
water levels should be discouraged or should be avoided; then salinity should be avoided
or salinity should be minimized by using the surface water, which is the fresh water into
the ground water; that means you dilute with the ground water with the surface water, so
that the salinity levels decreased. Then water logging, where the ground water level rises
to near surface, near ground surface, this should be avoided. So, these are some of the
objectives that we need to look at.
Now, whatever I just mentioned, this picture represents that. Let us say, this is the agriculture
area, in which we are growing certain types of crops, let say wheat or paddy or some other
crops like this maize and things that. Now, this crops here will have a root zone like
this; the effective root zone. By root zone, I mean the zone of soil below the crop, from
which the crops can extract water. So, in general, the root zone will be, the effective
root zone will be more than the physical root length that the crops have, because the crops
can suck water from this particular soil, which may be more than the physical length
of the root itself. Now, if you do not use the ground water correctly,
what may happen is the ground water table may keep on rising, and then enter the root
zone. So, this is the ground water table. So, it may enter the root zone. Now this causes
the problem of water logging. However, if will keep on lowering the ground water table
that means, if you keep on exploiting the ground water without properly recharging it;
then what happens is the ground water table goes so below the ground level that it becomes
unsustainable or unusable. So, this causes over drought of the ground water. So, we may
want to have the water level of the ground water table to fluctuate within certain band,
it has to below the root zone, and at the same time it should not go below certain specified
water levels. So, we may specify desired level of ground water or desired than within which
the ground water can fluctuate.
With that background now, we will see how we formulate simple systems models for conjunctive
use of surface and ground water. So typically, you will have a surface source, you may have
a several surface sources, but the surface source may consist of a surface reservoir,
there may be a dam with a canal network and so on. So, you have a surface reservoir; so
you will recover the surface water resource data, these will include reservoir details
for example, all this salient features of the reservoir, the area capacity relationship,
and height of the dam, the type of the dam, the type of canal network and so on. Then
you deal the hydrology at that point including the inflow, evaporation, seepage losses and
so on; the complete hydrology is what is required. Similarly, at the ground water, you may need
the type of aquifer, you may need the boundaries of aquifer, then aquifer parameters typically,
the storage coefficient time, the transmissibility, and you may need a calibrated ground water
model, because we are now talking about conjunctive use of surface as well as ground water, and
there is a continuous interaction, so ground water has its own flow mechanism, you need
to be able to model the ground water flow, and the surface water has its own hydrology,
and that hydrology needs to be model. Then you need to know the geologic condition,
because we are also talking about the recharge that adds to the ground water and therefore,
the geologic conditions are important. Then water distribution system both for surface
water as well as for the ground water, how many well, how many wells are there, what
type of wells, what kind, what location of wells and so on. And then of course, a water
use pattern; this should depends on the cropping pattern, and what type of crop is grown, what
is the crop calendar, and what is the evaporative demands, evapotransportive demands of the
crops and so on. So, all of these information would be necessary for getting a good model
on surface and ground water conjunctive use. Now, assuming that all of these information
is available, what we will now do is, we will take a simple case of one surface reservoir,
and one ground water reservoir, together we will see, how there is a interaction, and
then how we model this? I say it is a simple model, because we are not considering the
continuity of ground water storage with respect to the type of flow that takes place, where
looking at it as a box, in which the ground water storage is fluctuating. And similarly,
we put levels, within which the ground water levels can fluctuate and so on. So, it is
a lumped model, its its in a very large scale, it is a very large scale lump model, you can
replace it with any of the distributed models for example, you can take Morflow and then,
whatever I am saying now within the Morflow. So, you can include as many details as you
desire, and such modules are readily available, we should able to plug in those models and
then address the problems of conjunctive use. However, the focus here now is, how to formulate
the problems, given that there is a surface water model, given that there is a ground
water model, you need to put them in a system’s model that is a focus of this lecture now.
So, we will take a system like this, where you have a surface reservoir; at the surface
reservoir, there is a inflow, which is governed by the catchment of the reservoir, the inflow
in the catchment of the reservoir, and then there is an evaporation taking place, there
may be also direct seepage from the reservoir itself; you have a canal release, you may
have a canal network here, and this is the just a conceptual diagram, in which I am showing
just one canal, but it is the actually a network of canals; and the canal release is made to
an irrigated area, there is a seepage that takes place as the water flows from the reservoir
to the irrigated area; this is through the canals, and in conjunctive use modeling, perhaps
you will allow for some seepage rather than making everything perfectly lined and preventing
the seepage, because the idea is also partially recharge the ground water.
So, typically the field channels etcetera, you leaved and lined, so that there is seepage
that is taking place. So, canal seepage goes as canal recharge, part of it at least goes
as canal recharge, then at the irrigated area, you have the rainfall, and then you have excess
water going as depercolation that is the water above the field capacity of the soil goes
down as the depercolation; and part of this depercolation will add as recharge. You also
have pumping at the ground at the aquifer to supply water to the irrigated area; and
part of this pumped water also comes down as deep depercolation, and then it joins the
aquifer. At the aquifer you may have a boundary like this; you may have a net inflow and a
net outflow. So, the aquifer will take it as one box, and look at the continuity here;
irrigated area we take it at one box, and then look at soil moisture continuity if desired;
then the surface water reservoir is another box, in which we look at the surface water
continuity. So, simply put all of these different components
together, and then look at an optimization problem. What is the problem here? The problem
is to decide how much to be released from this reservoir here; and how much to be pumped
from the aquifer, during each intra season time periods. Now, these time periods can
be typically a month, it can be 10 day period or it can be a seasonal, seasonal time periods.
So, you decide on the time period of planning, and then you look at the optimal use of surface
as well as the ground water. So, the question that you are asking is, during each of this
time periods, you have so decided, how much should be released from the surface water,
and how much should be released from the ground water, such that over a period of time, the
system becomes sustainable; in the sense that, you are able to meet all of these demand;
yet at the same time, you are able to maintain the ground water levels within those prescribed
limit, such that it does not come nearer the surface nor is it overdrafted that is the
idea here.
So, the decision variables that I am taking about are the reservoir release during each
of the time periods. Now I am formulating this model, this particular example for monthly
time periods. So, we will do the operation for monthly time periods. Remember, this is
the planning model, where you will plan how much to use from surface water, and how much
to use from ground water; the actual operational model, which is the real time operation will
be slightly different, where you will take the input from the planning model, and then
incorporate in real time. So, right now, you look at these as a planning model that means,
how much to be used, how much do we plan to use form surface as well as from ground water
that is the idea here. So, R t is one of the decision variables,
that is reservoir release during time period t, and ground pumping during time period t,
GW t, so I will denote this as GW t. Now we will say, as the simplest first cut of objective
function, we will say that we want to maximize the net benefits in a year. Now the net benefits
need not be always economic benefits, you may have intangible benefits also included
in the model. But right now for simplicity, we will simply take it as economic benefits,
occurring out of the water resources system operation. Which means what? The amount of
water that we applied to the irrigated area in every time period t, both from surface
water as well as from ground water has a benefit associated with it. And this we will assume
that we are able to express in terms of the economic benefits. So, that is why, we say
that we would like to maximize the net benefits in a year, for its conjunctive use system.
Then on this, we will have to put the constraints. So, we define the objective function, we define
this decision variables; now we will there has a set of constraints that we need to look
at. First we have to look at the ground water balance that is from time period to time period,
we are taking monthly time periods now. So, starting with first time period, which is
the first month, we need to look at ground water balance; how the ground water levels
are fluctuating, which means we need to look at the initial storage, ground water storage
to which how much is recharged, how much recharge is added, how much of it is taken out as pumping,
and how much is loss through other means, which will discuss presently, and how much
is at the end of the period storage. So, that is the ground water balances.
Similarly at the surface water reservoir, you have the surface water balance. The inflow
comes in, there is a existing storage, and then there is a evaporation taking place,
there are other losses through seepage taking place, and then you are releasing certain
amount of water that may be spill, that is taking place at surface reservoir etcetera,
all of these together will govern the surface water balance. Then we may specify the maximum
and minimum draw down; that means, we will specify that the ground water level, water
table should be within certain certain band that is the maximum and minimum water draw
down. Then you may have storage limits at the reservoir that is the surface water reservoir,
you may specify a dead storage, and there is a capacity associated with it.
So, the surface storage should be within these limits that is reservoir storage limits. Then
you have the irrigation requirements, because you are operating this particular system for
a given cropping pattern, and you would like to meet the irrigated agriculture demands;
and therefore, you will specify the constraints associated with the demands themselves that
means, every time period you need to meet the demands to the best extent possible. Then
apart from specifying the limits for the ground water level, for the ground water fluctuation,
you may also want to place certain upper limits on total pumping through the year. So, you
may specify certain constraints on the ground water usage by specifying the volume of water
that is pumped from the ground water all through the year, which means every time period that
is the decision variable associated with the ground water pumping. So, you know how much
is pumped every time period, and you aggregate that the total pumping that you have decided
based on the model should be less than or equal to certain pre specified amount. So,
that is the another constraint, that you may want to put.
So, like this depending on the type of situation that you get, you may specify different constraints.
So, we will take these constraints, and then look at how to put it as a optimization model.
So, what did we do? We decided the decision variables, we set the decision variables or
the release from the reservoir surface reservoir R t during time period t, and the ground water
pumping during time period t, GW t; these are lumped models, so we are looking at the
total pumping from the ground aquifer, and the total release from the reservoir, surface
water reservoir. And then we decided on the objective function, we set the objective function
is to maximize the net benefits occurring out of the water application at the reservoir,
at the irrigation area. So, we are applying the water both from surface
source as well as from the ground source; the actual amount of water is that is applied
at the irrigated area has the net benefit associated with it; and then that is what
we want to maximize. And then we decided on a set of constraints including the ground
water level fluctuation, including the storage that fluctuate fluctuates between let us say
the dead storage and the maximum storage, including the meeting of irrigation demands,
so we may want to say that R t must be always greater than or equal to D t, where D t is
the demand, and R t is the release and so on.
So, this now, we will put more formally in a mathematical form. And look at how we formulate
the different constraints and the objective function. So, first we look at the ground
water balance itself. Remember we are talking about the ground water as one box here, in
which there is a net inflow and there is a or there is an inflow and there is a outflow,
so we may have a net outflow from this system. And this e has storage, and we may say that
the storage during every anytime period, must be within certain limits, and we assume that
we have a model, because of the parameters being known; from the storage, we are able
to convert the storage information into ground water level information.
So, we will be able to access both the storage as well as the ground water level at each
of the time periods. Now, how to do this is the different and slightly more involved topic
altogether, we will not go into this; for the purpose that we are looking at namely,
put it in our systems model, you assume that this is like a box, in which the storage is
fluctuating much the same way as the surface water reservoir storage fluctuates, and then
we will incorporate constraints associated with it. So, what we will do is, we will take
this is as the change in the storage; there is a net inflow that is coming here, and then
there is a out flow that is going in, going out of the system. So, we will take this as
change in the storage. So, we will say volume change in ground water
storage, this is S gw; we include all the terms that add to the ground water storage.
So, all these positive terms are actually adding to the storage, and all the negative
terms indicate how much is extracted, for because of various reasons from the storage;
we will see one of the one by one all of these terms, and then you may have a plus minus
term, depending on whether it is adding or it is being extracted. So, the left hand side
here is the volume change in ground water storage. So, change in storage. Now, the first
term W p is the recharge from precipitation infiltration; as I said from in the irrigated
area, there is a precipitation taking place, the precipitation in excess of the field capacity
in the soil that is the precipitation, that is going down into the soil, in excess of
the field capacity, which is infiltration in excess of the field capacity goes down
as recharge. We will assume that all of that goes down
recharge, although you must remember there is a unsaturated zone, water zone in which
the flow also takes place horizontally. So, not all of the seepage or not all of the infiltration
comes down as the recharge, but for the time being we will assume that all of that comes
as recharge. Then you may have recharge from streams, lakes and other natural water bodies
that is W r.
Then we have recharge by storage structures, canals, distributaries and other irrigation
works. So, these should because of the seepage through the canals, and the storage, and so
on. Then you may have the ground water inflow, if you are looking at the aquifer boundaries
like this, typically what we do is in the conjunctive use models, we look at the irrigated
area, which is on the surface, and then we demarked the boundaries of the irrigated area,
and then look at the aquifer boundary below the irrigated area. Physically the aquifer
boundaries and the irrigated area boundaries will not match with each other, but for the
purpose of conjunctive use we take only that much area of the aquifer or that boundary
of the aquifer, which is just below the irrigated area. And therefore, anything outside of that,
this is now looked at as a control volume, and then anything outside outside of that
you account for through inflow into that system as well as outflow from the system.
So, GW i is this, what is coming into the control volume or the aquifer, then you have
recharge from surface water applied for irrigation. So, this is W as star. Then W ag is recharge
from return circulation of ground water applied for irrigation; what we meant here is that
as you see from the surface water, there is a part of thing that is coming down, and adding
to the aquifer, which is the recharge due to the surface water application. From the
ground water, there is also a recharge that is taking place, and that is return circulation,
from the pumping that is what is meant by these two terms. Then you have GW b, which
is the ground water discharge to streams and springs. So, from the, within the aquifer,
you are also taking out some water or it is contributing some into the streams as well
as springs. Then you have GW e, which is the ground water
extraction by pumping and flowing wells. So, you you have actual pumping or some wells,
which are just directly flowing into well irrigated area, and then you have the ground
water outflow. So, these are various components; if you want to go into further details, you
can also add the ET losses of phreatophytic vegetation that means there are vegetations,
which are directly dependant on the water table itself; they take the water directly
from the water table. So, they are called as phreatophytic vegetation; typically, these
are large depths of roots, that means large lengths of roots, which extend into the water
table itself. Then you may have other terms included depending
on the situation for example, you may have artificial recharge through injection well,
directly it may be recharging into the ground water, in which case it becomes a positive
term. So, like this you identify various terms in the ground water balance; this is simply
the net chain in the storage is equal to what has been input minus what has been taken out
that is all.
Now, the recharge components that we are talking about two different recharges now, several
recharge in fact; one is directly though the rainfall, another is what is applied on the
irrigated area. This comes as infiltration, and whenever the moisture content below the
crop is more than the field capacity, the moisture content in excess of the field capacity
comes from as deperculation, and we are assuming that all of the deperculation goes as recharge.
So, the recharge component depends on first of all the geology, what type of aquifer that
you have; it may be hard rock or it may be alluvial and so on. So, it depends on the
geology itself. Then the intensity and duration of rainfall,
evapotranspiration, soil moisture, runoff, infiltration capacity, storage characteristics,
movement of ground water in the aquifer, and then the flow in the unsaturated zone that
is the that I just talked about, and so on. So, that are ways of computing the recharge,
which generally is based on all of these factors. We now going to these details, we will simply
assume that we know how to compute the recharge, for a given data, which pertains to all of
user match.
Then at the surface water reservoir, you have the surface water continuity, this is the
same continuity that we have been talking about in the reservoir systems. So, will write
the change in storage S t plus 1 minus S t is equal to, this is the inflow minus R t,
which is the decision variable minus E t, which is a evaporation and all other losses
together. So, we will write a simple continuity equation like this. Remember R t is the decision
variable here, S t also becomes decision variable, whereas I t and E t are the data.
So, simply what we are doing is for the surface water storage, as I mentioned the surface
water source need not be always a dam and associated reservoir, it may be simply a tank
or several tanks together, irrigation tanks. So, we are looking at irrigation tanks and
ground water usage simultaneously or conjunctively. So essentially, we are looking at the inflow
that is added to the storage that is the seepage that is taken place, you have an out flow,
and this R t is the decision variable, and then from this evaporation is taken place.
So, this decides the continuity or the storage continuity.
Now, in this broad framework now, we will take a simple example, I keep repeating simple,
because the conjunctive use problems are in fact very complex. You need to look at the
two dimensional ground water module, you you may have two incorporate or include two-dimensional
ground water model into the optimization problem itself. We will come to all those complexities,
if the time permits at the end of this lecture. So, from the draw down conditions, this for
an example, we will say that the total ground water that is pumped in the year is Q p. So,
we will specify the volume of ground water as Q p, then whatever is released from the
reservoir as it goes through the canal network, 30 percent of that goes as canal recharge;
that means it goes out as seepage through the canals, and then adds to the canal recharge.
So, 30 percent of the release goes as recharge. Then whatever is applied at the irrigated
area, 10 percent of the water applied comes down as recharge; now whatever is applied
at the irrigated area may come from surface water source as well as or from ground water
source or from the both these sources together. But whatever is applied, 10 percent of comes
of that water also comes as recharge. Then we have the inflow at the reservoir as I t;
and we will say that net benefits for each unit of water applied in period t as B t;
we will indicate this as B t; we will assume that monitory returns are known, per unit
of water applied in every time period. We will denote the ground water storage in
period t by GS t; so this is ground water storage. And we will specify that the GS t
should be within certain limits, minimum to maximum. Similarly at the reservoir, we have
the reservoir capacity, and the irrigation demands are denoted as D t. So, all of these
are data, we know I t is known, D t is known, and Q p is fixed. So, all of these is data;
this is known, this is known, this becomes the decision that is GS t becomes the decision
variable, as indeed is S t, which is the storage at the reservoir.
We will write the now the optimization problem. The objective from objective is to maximize
the net benefits, occurring out of the conjunctive use of surface as well as ground water. We
know that the benefit per unit water applied at the field is B t per unit water; and how
much we have actually applied? We have applied GW t, which is the through the ground water
source, and 0.7 R t which is from the surface water source. Remember we are talking about
the actual applications at the field level. So, this is the irrigated area, and you are
bringing R t here, 30 percent goes off as recharge. So, what is the actually applied
here is 0.7 R t; and at the same time you have also applied GW t, which is the ground
water pumping. So, this term here that is GW t as well as
0.7 R t, GW t plus 0.7 R t will indicate the total amount of water. And for every unit
of water that you apply, you have a benefit of B t therefore, this gives you the total
benefit occurring out of the water application. And this we sum over t equal to 1 to 12, there
are time periods, and therefore this gives you the total benefit across the year. Then
we write the reservoir storage continuity, and ground water balance; in the ground water
balance now, we have discarded several terms, we will simply take it as GS t plus 1, which
is the ground water storage at the beginning of time period t plus 1 is equal to the ground
water storage at the beginning of time period t plus whatever as come from the canals seepage
as recharge, we released R t from the reservoir 0.3 of R t that is the 30 percent of R t comes
as canal seepage, and that is the recharge. Then whatever is applied at the irrigated
area, we applied ground GW t, which the pumping from the ground water plus 70 percent of the
release R t; and this is the total amount that is applied; 10 percent of that comes
as a recharge, so that adds, and minus G W t; now this is the decision variable, G W
t is the pumping. So, these are the two major constraints that we need to look at; one is
the storage continuity, another is the, that is the reservoir storage as well as the ground
water balance.
Then we put all other conditions that ever ground water storage must be within this two
limits; the surface water storage must be less than or equal to S max. And then we put
the condition that whatever you apply at the irrigated area, namely the ground water pumping,
ground water application plus the surface water application must be greater than or
equal to the demand itself D t.
And we have also specified that the total pumping in the year must be less or than or
equal to Q p, we have said the Q p is the maximum pumping that is allowed in terms of
volume. So, every time period in time period t, you have GW t as the pumping. So, the total
in the time period, in the year is t is equal to 1 to 12 GW t, this should be less than
or equal to Q p; for this is not there. So, this is the maximum permissible pumping in
a year. And then because you are doing it in a deterministic sense, you will put S 13
is equal S 1, and G S 13 is equal to G S 1, to indicate that end of the period storage
is equal to beginning of the storage for next year. And then you also add the non negativity
conditions. So, this is the complete model, although a very simple model, I keep repeating
simple, because you it is all lumped, ground water aquifer is lumped, the irrigated area
is lumped, surface water reservoir including the canal networks everything is lumped.
So, this is the model that we write; and then we will take the example, where the data is
given like this. You have the inflows in million cubic meters, you have the demands in million
cubic meters, you have the evaporation in million cubic meters.
And then you also have the net economic returns. So, these are the B t values; now the B t
values are for several for 12 time periods, these are like this; 60, 50 etcetera, remember
B t is the net benefit that you get per unit water applied at the irrigated area. Then
reservoir capacity is known; then we also specify volume of water that can be pumped
from the aquifer over the year, which is Q p, Q p is known; and maximum volume of ground
water that is allowed to be pumped in a period. Now, these we may put it as G max that means,
GW is our decision on the pumping, but we will say that GW t, which is any period t,
the pumping in any period t must be less than or equal to 200 million cubic meters; you
should not pump more than this. Then we have taken canal seepage as 30 percent, and recharge
due to irrigation as 10 percent. So, all of these data are known.
So, we put this data into this model now. So, this is known, this GW t is being it comes
out as decision variable, R t comes out as decision variables, and in this, S t comes
out as decision variables, R t comes as decision variables, GW t is decision variables and
so on. So, all of the terms that are required to run this model are given now; through this
data, we use this data, and then make the run using a lingo model, it is a linear programming
problem. So, we use lingo, and then obtain the solution.
Now, the solution here is, it provides how much of ground water to be used, and how much
of release to be given from the reservoir. these are all in volume minutes, and for comparison
purpose of seeing, for seeing how much of the demand is met, we put this term now; this
term here is the water applied, actual water applied at the irrigated area. And this is
the water demand. And we also look at the recharge that is taking place during different
time periods. So, the demand here is not met for several time periods, and there is recharge
that is taking place, so that the demand can be met in the other time periods to the extent
possible. Now you can do several analyses with this.
So, this is the type of model, several types of analysis with this. First thing is you
see that there are certain periods, in which the demands are not met; for example, this
period, there is a big deficit; this period, there is a big deficit, and this period, there
is a big deficit and so on; which means that there is a crunch recourse both with ground
water as well as surface water together, there is still a deficit of resource. So, you can
start doing a sensitivity analysis, start increasing the reservoir storage, and then
see how the solution will be; start increasing the ground water availability, and then see
how the storage, how the solution changes and so on.
So, typically from such solution, you get a idea of what kind of demands that are met;
for example, you look at this red line here, this is the demand pattern; and anywhere else,
which is not seen, it coincides with the green level; green is demand required. So, wherever
red is not seen, it is coinciding with the green. And there are several time periods,
using this typically between 2 and 6 months as well as from 8 months to 12 months, the
demands are not met to a great extent. Then you will also look at how much of ground water
that is pumped, how much is released from the reservoir, and the demand that is required
and so on. So, you look at all of these in fact, you can also put the recharge, what
kind of recharge that is taken place and so on.
Now on this, you can do several sensitivity analysis to see how sensitive is the solution
with the reservoir capacity K, you try increasing the reservoir capacity K, but you will very
soon realize that this solution may become insensitivity, insensitive to reservoir capacity
itself, because it may be limited by the inflows. If your inflows themselves are small, then
no matter how large a dam you build, you may still be not able to meet the demand. And
therefore, you may have to start increasing the ground water that is available, but the
ground water will be limited by the actual aquifer conditions therefore, you may not
be physically able increase the storage beyond a certain point. So, this is how you model
the conjunctive use of surface as well as ground water, and this is the simple simplest
model.
You can also include a two-dimensional unsteady flow for the ground water flow itself. In
this example that I just discussed what we did was that we took took it as a lumped model.
However, you can go into two-dimensional unsteady flow model, and write a finite element code
for this convert this differential equation into a finite element numerical method; and
then at each element, you write the continuity equation. So, this type of complex models
are available in literature, you can just go through it, but the broad philosophy of
the system’s model remains to be broadly what I have discussed here namely, that you
identify a objective function, write constraints associated with the ground water, write constraints
associated with the surface water and so on. Now, if you if you want to put a finite element
model for the ground water, then all of these constraints that I have return must be written
for each of the element. Typically we may write element to be choose the element of
size 2 kilometer by 2 kilometers; and then write the constraints for each of the 2 kilometer
grid; and it will also depend on the soil moisture balance; it may depend also on the
soil measure balance at each of these nodes the elements that you have written, 2 kilometer
by 2 kilometer; and then the mesh type that you choose for the finite element and so on.
But such complex models are in fact, available. And then on the top of that you can also include
the uncertainties associated with the inflow, rainfall, the evapotranspiration and so on;
you can build stochastic models for conjunctive use of surface and ground water, but broadly
the philosophy remains the same. So, in today lecture then we have just discussed the conjunctive
use model; we will continue the discussion in the next class, where we introduce another
type of model formulation, not for conjunctive use, we will look at certain case studies
and so on. Thank you for your attention.