Lecture – 23: River Equilibrium-II
Good morning all of you today let us have the next lectures on river equilibriums in which we will talk about the next part of the river equilibriums is that will revisit the regime relationship. We will talk about downstream hydraulic geometry and river meandering and also the basic concept to the advanced levels. How the river equilibrium concept has evolved with time is what I will today Im going to present you.
So if you look at the next part it is very interesting that I have taken a new book concept which is a river training and protections work for railway bridges by Indian railways of institute of civil engineers which is published in 2016. Probably we will have some cross reference of this book about Indian knowledge about river equilibriums, river morphology that the concept river characterization regionalizations of river characterizations. These are the things are very detail is given in this book and I am going to follow this book as well as PY Julian books as you know it many of the study has been done for the university of many study has been done for the river Mississippi and lot of experience on the Mississippi rivers and elsewhere’s. The combinations of both I am going to deliver the lectures for you.
Now if you quit the concept if I liquid the river has a n number of degree of freedoms that
means we can visualize the rivers as the n number of degree of freedoms in terms of flow
variables in terms of sediment variables in terms of meandering channels the river bend it has
the variability. So how do we do it we start doing the the river survey so we look it to the
river survey which earlier used to go to the field conduct the river survey measure the
velocity depth width the perimeters of discharge sometimes may be the bed shear stress, bed
All we can sediment load, the bed load but we can measure it also they use the satellite
imagery to know it the bend characteristics in terms of the river bend length, meander width,
the radius of curvatures and what could be the tanλ the stream line deviations in the river
bend the sediment characteristics like ds the shield number sediment concentrations all either
you compute direct measured directly in the river or we can estimate from other data when
you have that.
So the definitely the river survey plays a major roles to find out the river the variables in
terms of flow variables sediment variables and the meander variables that that is what we get
it data and if you look at the in terms of degree of freedoms what are there in the flow that the
discharge, width, flow, depth, velocity, perimeters, slope and also the for the river bend you
can have a stream lines deviations.
For the sediment part we can have particle size distribute value shield number sedimentary
concentrations the meander length width radius of curvatures. So we can get a large numberof data if I do the rivers survey using go to the field collect the data or we can use a satellite
imagery to estimate some of these characteristics from satellite data. Nowadays you know it
it is quite easy to get the satellite data as a Google earth, Mobas or Google earth is there and
many others satellite data providers are giving a free data set to characterize in the river in
Now if you look at that what did they do it the basically targeting the regionalization’s can
we establish the relationship between independent dependent variables. Anyway you can
expect it that the independent variable like discharge, the dx and the shield numbers can be
the independent more than that we do not know it so that way if you look it by just looking
this data so discharge which is easy to measure it.
And you can have a ds value you can have a τ* the shield numbers and we can try to locate
whether we can establish the relations with other dependent variable like flow width, depth,
the velocity, the perimeters or the sediment concentrations or we are looking at it what could
be the meander plan form in terms of radius of curvature in terms of the length width whether
we can have established the relationship there.
And the questions comes that can you obtain a statistically significant relationship among the
flow variable, sediment variable, meander dimensions at a regional or the global scales. If I
talk about the regional scale can you develop the relationship for the Ganga, Brahmaputra
meghana systems. Can you develop the relationship for the Peninsula Rivers or can you
develop a relationship which can hold good for at the global scales.
So all these bigger questions mark people lot of research has been going on to find out
whether there is no what you generally do it there are regionalizations if I summarize that the
basically it is a data mining concept. This is data mining concept people nowadays talk so
much. So data mining concept you have a huge data measurement of the flow variable
sediment variable the measuring characteristics at different part of the rivers different rivers
at different time intervals at the equilibrium stage.
The stage where the banks are quite stable or the there is not a significant order of change is
happening it. If you look at that part we can establish the relationship using the data mining
concept very basic is correlations analysis visual 3d plots that a lot of data mining algorithmsnowadays available and those things we can integrate it and evaluate the relationship. What it
happens? What is the known empirical relationship with us? Lacey's equations we will talk
about that lane’s equations 1955, 1929 Schumm 1963 all these relationship I will talk about.
Leopold relationship 1964, so these are the relationships we got it after doing a data mining
of huge data what we collected from the river level at the reach levels at the different part of
the world. And we try to integrate it either the regional scale or the global scale try to look it
whether we can establish empirical relationship between dependent variables and the
independent variables of a river systems.
If you look at that what we try to look at that this river is behaves differently there has a
concept of equilibrium concept is there so it is a lane’s concept or the lacey’s equations
concept will come. Variational problems also there they consider minimize the energy
dissipation concept follows through the variations is the Julian 1985. The large scale eddies
concept nowadays people are talking about all these forces what it happens it that due to the
extreme flood events it creates a large scales eddies.
The turbulent structures what we discuss previous classes that that structure is responsible to
have a formations of that. So that is the reasons there are the large scale eddy concept which
is Yalin and De Sliva. Recently we also introduced an entropy concept for the river
morphology which you can follow it and some of the papers also we will discuss in the next
class that how we are we have brought it a concept of entropy into the river morphology
analysis that what we will discuss it.
But if you look it as look at as a summary what we are understanding that we need to have a
river survey we should have a quality river survey data that is what is matter to us if data is
not a having a quality data we can land up with empirical equations which are not statistically
significant. We should identify the variables flow variables, sediment variables, mandarin
characteristics and much more we can look it.
Recently there are a lot of data mining algorithms are there and the correlations visual 3
dimensional plots which are there nowadays any spread sheets Microsoft spreadsheet and you
can establish the empirical relationship what we got in almost 70 years back those things wecan look it but there are new concepts which is are coming it starting from equilibrium to
entropy. That is the basic idea when you do a empirically relationship.
So try to understand it the empirical relationship of a river systems talks about the river
behaviors which we do not understand at the particular time scale or the spatial scale that the
idea is if it is there is a empirical equations. Let us I repeat it lacey's equations again I want to
revise this lacey’s equations for you to because this is the equations we so frequently we have
been using it river protections work.
(Refer Slide Time: 11:27)
So lacey’s equations if you look at that it identified the basically the velocities area the
hydraulic radius and the P which are dependent variables is a functions of discharge and silt
factors. The silt factor is a function of particle size d50 value functions of particle size
distribution. So this is a silt factors and these are all conducting a series of experimental data
from the canals in Indian subcontinent undivided Indian subcontinent.
So if you look at that if you look at this lacey’s formulates what you widely use it it has still
has a strength like manning’s equations it has represent the lumped response of a equilibrium
reach of the river or the straight reach of the rivers where the dependent variables like the
velocity, hydraulic radius, area, perimeter and the slope is a functional of Q and f 1. Q is
discharged f 1 is the silt factor it is just a data mining concept.
The equation has not come up just like that they have done thorough analysis of the data
using way back in 1929's okay we did not have a much computer that times. But is that dataanalysis concept the same concept we are using it with super computers now but there is a
data analysis concept where it has established the relationship between dependent variables
and the independent variables.
Independent variables here define is about the sediment transport the d50 values and the
discharge the Q representations and that what you have the velocities hydraulic radius, area
and the parameters. So just you look it, the velocities depends upon for equilibrium channels
is a Q as a power relationship with 1/6 and there is a factors f 1 factors are there which is 1/3.
Same way if you look at that hydraulic radius has this coefficients.
If you look at this most of the coefficients are simplified in terms of 1/6, 1/3, 5/3 or 1/2 that is
basically the perimeters representing Q ½ it is too easy to remember it the p is equal to
2.66Q1/2 its very easy. If you look at this equations what we rewrite it the P is equal to the
perimeters is equal to wetted perimeters is equal to 2.66 times of this square root of this
So that means it is just depends upon the flow. So the perimeter of the river wetted perimeter
of f equilibrium channels it depends upon your only the discharge value only the discharge
value. So if you look at that and that what is a power of 1/2.5 square root of this. If this very
wide river the P will be close to the width of the rivers so that means for a wide river like a
Brahmaputra or Ganges the width is a just a functions of the discharge that is all.
It does not depend upon the particle size. So how can it happen it but it is the relationship
after doing the data mining. We do know it because these are the data the canal level data is
collected from the field levels then done the statistical analysis like the meanderings equation
the Lacey’s equation has a lot of strength. And if you look at that part the same way for the
velocity, the hydraulic radius will be for a wide river it will be close to the flow depth.
So that means the flow depth has a function of Q and f and this having functions of Q 1 by 3
and f 1 -1 by 3. See if you try to understand it how these functions has come up and these are
data mining after doing it so called todays we talk about data mining concept similar way
they collected data they did the outlier analysis they did visually all these equation fitting and
then they try to bring it the relationship which might have taken few years to bring thisequation it is not a few days like what we have now computing facilities with Microsoft excel
spreadsheet or any statistical tools.
There was no tools no computers but how they derive this concept through the data mining
collecting relevant data, conducting all these things and this is still holds good still in India
we can see the regional equations it is quite valid for the Ganges, the Brahmaputra alluvial
systems we use this equations as it being develop for a regional scale. Let we will discuss
much more about this when you go for next levels of regional equations.
(Refer Slide Time: 18:02)
Now if you look at the next part when how you define it in the bank full discharge. If you
look at that you can have rivers like this where you have the plodding and that bank is not at
all having any levees. But there will be the conditions that levee you within both the banks
because the water will flow it during the floods both the banks. If it does not flow those both
the banks then you will have outer bank levees or if I have a levees on the both the banks.
This is a natural levee natural levees processes are there no levees natural levees forces in the
both the banks and the outer bank levees. So if these the conditions now we have to look at
what is the bank full discharge, the lateral levees resulted from depositions of course of the
suspended loads on the floodplain adjacent to the channels that is what it happens it when
extreme flood events comes the flood of 2 years, 5 years return period now most of the water
goes to the flood plain area.As it goes to the flood plain area thus also it goes with the sediment loads and that sediment
start the depositing. And because of that depositions it creates a natural embankment that is
what is called natural levees it could be a both on the bank or outer bank or there are no
levees for that matter. Now if you look at that we also call about dominate discharge because
the single discharge may not represent it.
That it depends upon sufficient magnitudes and frequency to determine the dimension
geometry of alluvial channels just trying to repeat it. The dominant discharge is that the
sufficient magnitudes and frequency the occurrence to determine the dimension and geometry
of alluvial channels in this lecture. So we are not going more detail about dominate this chats
but the hydraulic geometry relationship the dominant discharge can do bank full discharge
which is having a return periods about 1.5 years that is what is there.
So that means the bank full discharge by conducting a thorough analysis is found to be a 1.5
return period flood discharge what does it indicate? It indicates is that if it is a bank full
discharge this happens in the return period of 1.5 years that every 2 years we are supposed to
be in a natural rivers we should have flood in the floodplain. Every 2 years intervals you
should have that is nature of the flow.
We should have the floods on the flood plains so when you do the embankments you try to
understand what we have been doing it because when confining the rivers. And as we
confined the rivers you know that how the sediments happen but naturally the river creates its
flood plains and flood plain inundations it happens at a rate of a 2 year return periods that
means every 2 years if we are living in the flood plain if a river as it in natural conditions.
We should expect 1 flood event. So if you look at that that is what the natural conditions but
as we modify the condition that is what we should try to understand what we are doing it. But
naturally what it shows that the bank will discharge is equal to 1.5 return period flows that
means 2 year return period flow, flood in a rivers it is supposed to inundate the flood plain
area supposed to inundate the flood plain area 3 years 5 years and 20 years no doubt it will
indicate a larger area.
So this concept can help us to understand how things are happening it at the rivers scales and
that is what is a regionalization idea that it has a return period of 1.5 years but no doubt weshould do regional studies to find out whether it is a 1.5 years or 2 years for Brahmaputra
river or Ganges or the peninsula rivers those research studies to be done it to look at what is
happening it whether it is 1.5 year return period that is for us.
(Refer Slide Time: 23:15)
Now if you look it if I go for next levels we try to look at the relationship the basic
relationship again we look at a simple continuity equations and the flow resistance equations.
If you look at this simple continuity equations which is a discharge is area into the velocity
here the W is a width W is width of the river and h stands for a flow depth V is the velocity
then you will have the discharge due to W h P.
So area into velocity but if you look at the flow resistance equations which takes care of all
the near boundary nearbed turbulent structures and all we can establish a relationship with a
velocity with h by d s so that means d s indicating for us if it is a river bed these are the river
bed it is the flow depth and d s is the particle size if you look at the stones or boulders are
there for hilly rivers. You are looking a submerged relative submergence h by dx to the
power exponent of m which also depend upon h by d s.
It depends upon the relationship between this with the whether it is the submerged how what
is that it is h is much, much larger than d s this values will be the larger value is a less you
will have these things. Just you look at the relative submergences value indicating first. Then
you have the slope you have a flow depth then you have the coefficients indicating for us the
resistance flow.So we have a continuity equations we have the assuming the velocities going on as a flow
resistance equations you can have this c and this power exponent m is depends upon relative
submergence. Banks are the non cohesive materials which is too easy to have a
simplifications no alluvial rivers is full of the sand, it is a mixture of silt clay and
compositions of that.
But if you are considered a non cohesive materials as we discussed earlier we can define a
shield numbers which are functions of 2 components 1 is the downstream shear force and
weight of the fluid sediment particles that is what is the shield numbers or shield shear stress
we talk about that is what we discussed earlier.
(Refer Slide Time: 26:21)
Now we try to have a relationship that is what is look at that you know wait these things the
particles on the wetted perimeters of algebra channels if you see that if τ* is less than critical
shear stress if it is more than that then you will have a have motions it will start entering the
motions and the rate of sediment transport increase with the shield numbers. The shield
number primarily depends upon the flow depth.
Because d s does not change it the g does not change it so specific gravity does not change it
only this h and s value so that is the reasons it is primarily depends upon the flow depth and
vertical process of the aggregation degradations in alluvial channels. But if you flow in a
bends as we discussed earlier there will be centrifugal forces there will be stream line
deviations upper stream line deviations will be there.And those things if you would relate the relationship you will get it again h/ds somewhat
relative submerged depth h/W, W is a width h is the flow depth. So that ratio and it has a
depending upon the br ratio which is a constant for us. So if you look at that how the flow
variables are coming. So if you look at all these things the dependent variables primary
dependent variable is considered is discharge d s representing of the bed settlements and the
τ* the shield numbers indicating that a ratio between the 2 forces.
One is a shear forces by the weight of the sediment particles. So these 3 consider as
independent variables and others we are talking about and the Julian 1988 establishes that
variability of other parameters relatively small that is the they are data analysis and data their
data mining concept what they got it these are the 3 are primary independent variable for all
them to establish relations with the dependent variables.
(Refer Slide Time: 28:53)
Now if you look at the next part it is a quite interesting which is given by Julian and
Wargadalam1995 the relationship between flow depth, width, the velocity and slope is a
functions of now it is a higher versions than the lacey’s equations is a functions of Q ds τ*, Q
ds and τ* it is a functions and the power exponent is depend upon the m value. So m is a
resistance coefficient which also have a relationship with relative submergence depth.
So as compared to the lacey equations now we are going more details which is now not only
depends upon only the Q and d s it depends upon the shield numbers it also depends upon
your power x point like in earlier case we have a width is equal to 2.66 Q into half. So widthof the river wave but in this case the width of the rivers has a (2m + 1), (3m + 2) plus there is
a functions of dx and τ*.
As we do a data mining advance data mining concept we suppose refine the equations we get
it more independent parameters as compared to the lacey’s equations. So if you look at that
we have this part which is are indicating for us and s is a friction slope what we have or the
energy gradient. And the flow velocity Q is a dominant discharge in cumecs, d s is a d50 in
meters and shield numbers τ* and the resistance that it is.
So what they did it they established these equations then they have a relationship between
measure and predicted. So that is the reasons when you develop the original scale things you
need to have a huge data set that is what is the statistical analysis. If you look it this is a
measured one versus predict for the flow width. So if it is follow in a 1 is to 1 line it is a
perfect but it does not happen for a river database is predictions and the measures should not
have a 1 is to 1.
Because that is the natural variability is there but their equations is perfect and that is what is
they have the prediction and the width in meters with from 1 meter to 1000 meters it is 1
meter river width to go to 1000 meters still you can have a relationship of width in terms of
discharge d s and τ* and it can follow these functions and it also showing this statistical
The same way flow depth which varies from 10-2 to 102 whenever you see the data in a river
case please try to interfere then you can have a interesting knowledge about the rivers these
are rivers speaks out the truth through this data. So we should interpret the data more
extensively look at the range. The major is at 10-2 which is a centimeter a level goes to the
100 millimeter and most of the rivers are in this range.
Most of the rivers are up to 10 meters or the 12 meters not more than that and that data set it
is false is here and which is a predicted range and that is the presence. So with this the a way
Julian Wargadalam relationship in terms of flow depth, width and velocity and friction slope
we can; the velocity has a much more variability’s which is a meter per second it goes up to
10 m/s that is what is maximum velocity can happen it ok.This is a 1 meter per second and it can have a 4 meter, 5 meter per second. So that is what I I
said it earlier the maximum velocity you can design for a reward training works we look
interpreting this data we can consider from this data as its measure you can have a roughly 5
meter for seconds. Because we design for a stream because many of the times in a rivers
system we do not have the data and we cannot wait for another 30 years to collect the data
measure the velocity.
But we can have interpret the knowledge about the rivers from elsewhere as a regional
studies like as its indicating here the river velocity which we need it many of the times it can
go as high its 5 meter per second or 7 meter per second not more than that as the data
recording is so great because we look for maximum. And similar way the flow depth also in
this and if you look at that way the slope also if you look at the predicted and these things
which is more statically significant band its follow it as compared to velocity.
So it all says that how much of uncertainty is there from the scatter plot when you try to
establish this relationship the original relationship in terms of flow depth, width, velocity,
energy slope with respect to Q ds and τ* and these are the data is collected at the river levels
at the field levels and with their comparisons with a major and predicts. So does that how
good the relationship if you look in these things you can say that the velocity can have a lot
of uncertainty in measurements and the predictors.
If you look at this equation as well as the measurement squared but in terms of depth and
width and the slope we can have a not much that significant order of uncertainty is involved.
(Refer Slide Time: 35:46)Now if you look it other very interesting things are there if you look at this channel width this
is very interesting figures if I plot it this discharge versus channel width which varies meter
10 -1 its almost a centimeter levels the width can go to 105 that means almost a 10 kilometer
width like Brahmaputra rivers 10 kilometer width and it can have a one kilometers and it can
go to the 10 kilometers width.
So if you look at that part what is indicating your discharge if you look at that just look at the
distance 104 so that means discharge can go up to 10000 cumecs this is very good in
relationship and we can interpret a lot of things these are the data at the river compiled all
over the world plotting between channel width and the discharge. So if you look at that if the
width of the river is 1 kilometers it can go as high as 10000 cumecs which is normal for
And if a width is increasing it up to 10 kilometers the discharge can go as a level of 10 times
that means 0.1 millions or 1 lakh cumecs that is the strength of the Brahmaputra rivers as we
can see the width of the rivers. And you just see the width of the rivers and we can know it
approachment range of the discharge because these are the data is plotted for original river.
So if I have a 10 meter width you can say that you can have a n log scale with the 10 meter
cube per second. So these are very interesting data which is compiled in 1983 by Kellerhals
and Church for a rivers specialist the width of the river we can measure from satellite data
and we can know it what could be the average discharge is flowing through that is this plotwe can use it to know it estimated discharge ranges that is what is a scale as I intentionally
representing you try to interpret this date the discharge versus the channel width.
As its width varies from centimeters to go to 10 kilometers or 100 kilometers which is not
possible so maximum range is 10 kilometers more than lesser than that and you have a
discharge with the 105 cumecs 1 lakh meter cube per second so Brahmaputra discharge is
72000 meter cube maximum highest 100 year return periodic time is much larger than that.
So if you look at that this data across the global level it indicates for us to interpret the
knowledge the relationship between channel width and the discharge. That is what if you
look at that the average flow depth is h in meter surface width W and if you have average
velocity v equilibrium slope if I establish is a functions of Q ds τ* and if you use the m equal
to 1/6 using this manning’s equations.
These equations again you can modify it you will get a relationship approximate relationship
between flow depth width velocity and the energy slopes and the energy slope which is a
functions of that. But just tend to have a representing you that if you look at that the lacey’s
equations talk about the width is a functions of a constant and functions of 1/2 that is power
function of 0.5 the same thing here is a 0.53 no doubt.
We improve the equations but more or less the power exponent of this relationship it remains
more or less the same that is the things you can try to understand it with the different things.
So in time we are evolving regional river relationship but more or less the power experts are
within the ranges that is what is the lacey’s equations is 0.5 and where is the Julian equations
is 0.53 is not significant difference. But there is a τ* and ds is there which may have a slight
bit difference is there.
(Refer Slide Time: 41:01)Now if you look at the next level to interpreted it or many of the times we if you know the
discharge and d50 value and we know the shield numbers at beginning of the motions can we
compute the downstream hydraulic geometry. That means we need to compute depth, width,
the perimeters and the velocity and the slope. This side is known d50 is known to us the
problem is here m is a function of dx and the h m is a functions of relative submergence
depth which is a function of h/dx.