 Hello everyone, welcome to the NPTEL course on groundwater hydrology and management. This is week four lecture four. In this week we are continuing to discuss the important components of groundwater hydrology and to continue our discussion we will look at hydraulic conductivity which is one of the most important parameters for groundwater hydrology. In the previous lecture we discussed about the macroscopic view and the microscopic view of groundwater flow and how the water interacts and moves according to the material the soil or the rock material. So in the macroscopic or which is called the Darcian view you have a average linear velocity of the water particles because in a discharge queue which goes around a cross-section area A the macroscopic view does not capture what is happening inside. So how does water move inside is not captured and not needed to be honest because in a macroscopic view it is averaged. Okay so you have an average linear velocity so the velocity is assumed the same on the top middle and bottom portions of the soil. But in the real world which is a microscopic view you see on the right hand side you understand that the water doesn't go in a straight line as we see here in the macroscopic then water would go in variable velocities along the soil medium and this is because of the heterogeneous composition and arrangement of the soil particles. So you need to understand that there are microscopic differences and variable velocities. However when you want to describe them in a particular equation you need a macroscopic view. So that's what Darci took over and he modeled this groundwater flow through a pipe. So he didn't want to do it on the land and ground because of the microscopic variations he did it in a lab setting. So Darci found experimentally that the discharge queue which goes into the soil medium is proportional to the difference of the height of the water h which is the hydraulic head and the difference we also call as gradient between the ends and inversely proportional to the flow length which is the length of the soil column through which the water flows. So you have q is directly proportional to h a minus h b we will see what h a and h b is in the experimental setup and q is inversely proportional to the length of the soil column. Before that just to introduce Darci Darci was an engineer in France where he worked on fountains water fountains and he had to supply water through underground to make sure that the water fountains were working properly. For that he didn't have any equation. This was very very old times and so what he did is he took the soil column to his lab and then did this equation and till date it is one of the most accurate very simple but most accurate description of the ground water flow in a saturated system. The assumption is also that the soil inside is fully saturated. We need to understand Darci's approach because Darci was doing this to send water into the fountain and the fountain needed a continuous supply of water for which the soil and everything was saturated and through a pipe column. So we can look at the differences in his experimental setup and how he got it this slide. So in the Darci's experiment what we notice is that the column of sand stopper at each hand so either sand soil rocks materials the material the matrix is kept in a tube and it is stopper a rubber stopper is put on the top and bottom okay. Water saturates the pores water is fully saturated inside the pore space. Now apply yourself to the previous lectures. If we say porous space is fully filled with water that means it is a saturated system and also it means there's no air inside the soil. So this column of soil he took in a tube has full water inside along with the soil particles. How do you establish it? You just continue to pour water until for a long time until the water comes out in the same rate okay. So q is the input rate and if the q water comes out that means full it is saturated. Constant volumetric rate of inflow and outflow of water q so he also maintained q to be constant coming in and coming out and that is what was needed to operate the fountain systems. So he would send a known volume of water which is q which is already measured and meter through a soil column of length del l okay or just l and then he had a stopper and q coming out. The cross section of the tube was measured as a and he had two points of monitoring inside the tube and those can be visualized as wells okay. So he wanted to know what is the flow inside the medium okay. So to understand the flow he put two monitoring points and between the points he's going to calculate the well hydraulics and the hydrology groundwater hydrology movement. So what he did is the first well is called well a let's say a and then the well b is in a lower elevation and why does water move from top to bottom because water flows from high potential to low potential. There's no other fancy instruments here just q he's sending in he doesn't have to push it because gravity is pulling the water from high potential to low potential. So when you put in the well automatically the water level in the well will equilibrate based on the atmospheric pressure the pressure inside should balance the pressure outside and it falls into a particular level after it equilibrates and that is h a we saw in the previous equation h b is the same water level measured in the well b okay. So del h or the hydraulic gradient it would be your difference in the h and divided by the length del l okay. So what we have here is a distance between the wells and also a difference in the head. Also what we need is the elevation of the well inlet from the datum which is zero. So here the table as zero and from there how much is the elevation of hydraulic head one which is h a and hydraulic head b which is at h b. So he measured the elevation of the well z one and z two then he measured the total hydraulic head which is h two and h one. So h two and h one are going from the datum which is the zero elevation. So from the zero elevation you go up to the water level this includes the elevation of the well okay the elevation of the well is the point at which the well opening is there from the ground from the ground which is zero and the hydraulic head is on addition to that the water column height is added. So now we have the total potential okay. So h one is higher than h two and that is why groundwater will flow from h one to h two because high potential low potential okay. So it flows from high potential low potential. So q is given as the volumetric flow rate a is the cross-sectional area h is the hydraulic head which is including the elevation okay elevation of the well from the ground and l is position coordinate in flow direction okay. So how much in the flow direction at length okay. So Darcy's law is given as the first law what it says it is proportional to the hydraulic head difference which is del h del h is the difference between h two h one and h two okay and also proportional to the cross-sectional area directly proportional. It is inversely proportional to the length between the wells between the estimation points. So now any proportionality can be converted to an equation by introducing a proportionality constant right. So this proportionality symbol would go to an equal when we introduce a proportionality constant and that constant is called hydraulic head conductivity okay. So the hydraulic conductivity is given as k and a is the area del h by del l. So if you take the area out I'm not interested in the total volume but I want the velocity or the discharge and that discharge q is rated by the area. So if you divide by area on both sides you get small q which is equal to minus k d h by del okay. So this is a very simple equation but captures the flow of water between two wells and also the discharge rate. Now let's take a step backwards why do we have a minus where did the minus sign come right. So when we discussed it is like okay it is proportional and it is inversely proportional to the length but where did the minus sign come. The minus sign is introduced by Darcy to document that the flow is in the reducing direction of potential. So the negative is a directional value not a value that you put on discharge you cannot minus the discharge okay. So what do you mean by minus discharge there's nothing as minus discharge but the negative sign indicates that the flow is in the direction of reducing head okay. So it flows from high head to low head and that capture that needs to be captured otherwise how do you know which side is the water flowing right. So to document that Darcy had introduced the minus sign and it flows from high potential to low potential or towards the decreasing head the head is decreasing. So all of this is captured in a very very simple equation and to be honest working with groundwater systems for the last 10 to 15 years I've noticed that this equation has been most powerful even compared to the newer equations that come now in that. So there's not much additions done to this equation and all these groundwater models are based on this equation for saturated flow. These equations and descriptions can be taken from Fries and Cherry book 1979 as I said it is one of the very important books that groundwater hydrologist would have and all these government schemes are also referring to these books. So this is the direction of decreasing head so the negative sign please understand it is a direction value not a value that you put on the quantity it is not a decreasing discharge minus discharge plus discharge it is the direction flows in the direction of decreasing head. Also understand the difference between a large q and small q large q is the flow which is a volume and q meter cube or l times 3 on the power but your small q is a rate velocity kind of thing so it is a discharge and l by t so the time would come in this equation whereas in the discharge it is volume per unit time. Moving on we have a horizontal pipe filled with sand to demonstrate Darcy's experiment in a different book I've taken a different approach to just show the same explanation but in a more different way than the Darcy's book. So horizontal pipe filled with sand so there it was slanting but here it is a horizontal pipe to demonstrate Darcy's experiment so as the book says it is originally vertically or slanted position but we can also explain it in a horizontal way. Why do we need to explain it in a horizontal way is to capture the ground system most of our ground is not going to be always like this in the field. So for example in a rural area you want to monitor the ground water flow between two blocks and the blocks are going to be straight it's not going to be like tilted etc but inside the ground it may be tilted and that is because of the layering you remember the aquifers we talked about so the layers might be tilted or different and that causes the head difference. So what this diagram says is that there are two wells and two wells have different head hydraulic head so HA is the hydraulic head on well A and HB is the hydraulic head on level B well B. We cannot ask why is one higher than the other it could be different aquifers it could be different pumping regimes so even though it's horizontal the levels are different now the L is taken as the difference between the wells right and so the basic equation is Q is equals to minus K hydraulic conductivity A which is the cross sectional area times DH by DL DH is your change in the head or difference between the head from HA and HB and your DL is the distance between the wells. Now this DH by DL is called the hydraulic gradient and what you also notice is that the K the property K is the one which captures the property of the soil because area A is basically the area of the cross section H is the hydraulic head which is a water property and DL is the difference between the wells. So where does the property of the soil or the solid come it comes in K hydraulic conductivity. So the conductivity as I mentioned in the previous class is a term given to the soil and how easy it conducts or lets the water to pass through and so we call it hydraulic conductivity which is a property or a function of the soil. Moving on please understand there are lots and lots of units for this values it can be expressed as gallons per day per meter square or meters per day it's all in a L by T dimension which is length by time so if you divide the gallons which is a volumetric term by area you get one length and the day is your time so you can have multiple units so please be careful with the units I recently was in an examination where the student was trying to say no it doesn't agree the data doesn't agree but then the simple thing was that the units were different so make sure the units are captured correctly I am going to stress this again and again please make sure what the unit is and see if it is agreeing when you compare okay so let's look at some of the hydraulic conductivities in the previous lectures we looked at permeability porosity differences and now hydraulic conductivity if you look at it hydraulic conductivity is a function or can be expressed as a permeability and along with gravity constant and your viscosity and your density of the fluid so it is basically a related variable so for clay it is a very very slow centimeters per second so think about 10 minus 9 centimeters per second is the flow rate kind of velocity of water in clay or how clay allows water to flow so it's very very slow so when you go to a field the first question you can ask is what peko soil it is and if they say clay then you would understand that a groundwater recharge would take long long time we'll have some examples in the next class so moving on silt and sandy sills the mixture of clay sand and and particles have a slightly reduced hydraulic negativity or a higher hydraulic conductivity because sand actually can improve the hydraulic conductivity so then when you go to silt sand and fine sand you have more hydraulic conductivity so we're going in increasing hydraulic conductivity okay and then when you go to well sorted sands glacial outwash which is the deposit by snowmelt and well sorted is that the particles are well sorted good space in between then your hydraulic conductivity increases so the highest would be at least on this data set it is the well sorted gravel gravels are bigger in grain size and they have well good built structure because it is well sorted and it has space in between so when you sort something it will have a lot of space in between and that space can make the medium conductive and that conductive will increase the hydraulic conductivity of the material so you would see higher water flowing through recharge and groundwater discharge through well sorted gravel and the least in clay clay also swells it also takes the water and holds on to it so all this is kind of captured by your hydraulic conductivity. We can also use a Fries and Cherries book that we used in the previous lectures for permeability let's take an example of again the silt sand because that equates to your fracture igneous metamorphic rocks if you go back to the class notes you understand that is the dominant aquifer system in India the hard rock aquifer covers more than 60 percent area and so if you take that we're going to take that as an example to understand hydraulic we'll use a centimeter per second or meter per second to discuss the results so you see that it is anywhere between 10 minus 4 to 10 minus 3 centimeters per second it's very slow whereas in your aquifers alluvial aquifers which is around 30 percent in the country the nganges ingus bramaputra the kaveri delta so those regions will have a higher hydraulic conductivity so those would be more on your permeable basalt or your clean sand the sand alluvial sand and then you could see jumps up the hydraulic conductivity can jump up from 10 power minus 4 to 10 power minus 1 if i draw a line here it is 10 power minus 1 centimeters per second so it is moving at 1 millimeter per second which is pretty fast okay so you're thinking about per second okay per day so then you convert it to per hour per day per year always go in time in sequence we'll do some calculations in the next class to show what does these value mean okay but before that i would like to also introduce the hydraulic head concept in this lecture so always note the ranges the range is big and why is the range big for example in silty sand the range is from 10 power minus 5 to 10 power minus 1 so minus 10 power minus 4 orders of difference and that is because of the mixture the combination of your silty sand could be different and also the management of the land could be different but either way it falls within this bank also note that when i just take one value let's say 10 power minus 4 it could be a clean sand on the border of clean sand it could be a silty sand it could be a silt lowest or a glacial tin all these four five different types can be within that's one value so it is up to you to understand first what that material is by by physical and lab estimations and then you pick your hydraulic conductivity water levels this is a very very important concept refer to understand the hydraulic conductivity and groundwater hydrology because that is what we're measuring okay at the end of the day the government and the system would measure water levels and i'm taking a small example just an introduction we'll jump into one lecture on this data to understand this so all this data would convert to a groundwater hydrology equation by the examples we showed in Darcy so this is a picture of the groundwater data that is calculated and captured by the government of Tamil Nadu and on your right is the central groundwater boat so there are two agencies at least in Tamil Nadu collecting groundwater data so they collect the data and then they collect it in different years and months and then they equate the difference to understand what is happening so before that let's get into the hydraulic head concept i've already explained this in the Darcy equation but let's take one well to show how do you calculate the elevation because in the real world the zero is not the table because the real zero is your center of your location where the level is zero so normally the zero is taken as the sea level on the planet the elevation zero is taken as mean sea level okay so where the sea is starting that is called zero and from there the elevation can go up or down okay so let's take z as zero which is your sea level we are somewhere in let's say Chennai and i'm on the beach that level the water level is zero and i'm moving inward into Chennai to measure the groundwater level i need to measure the water level the hydraulic head and from the hydraulic head i will go to q which is your Darcy's equation okay so i go to a well and the first thing you would notice is that the elevation of the well is not there you cannot have the elevation of the well readily the hydraulic head readily calculated so you need to calculate the first step you do is to ask what is the depth of the well and where the well is actually open for measurement so here is the point of measurement and as usual as i showed in the previous slide you would measure the water level or depth to the water so you end up calculating this so you have one value which is the depth of the well and also you have your water level which is side okay and what is needed for your q is h how do you calculate h you ask for the elevation of your ground surface at that point which you can take from topographic maps or digital elevation maps so you have the elevation of this your ground which is one okay now two you have this z okay which is your depth basically your depth of your well which is two and you know your side how much is your side which is the water level which is three but how do you get at two how do you know how much is my elevation from the ground you know from the depth of the well which is this you know okay this is your two and but you won't know this one because you have to calculate this indirectly okay so you know one you know two and three how do you get at h by subtracting two from one you get this area this length okay so you get this length when you subtract two from one so one minus two is going to give you your z okay and you know side which is your three so z plus your three will give your your z plus your three will give you your h and that's simple please understand that this is what we want but in order to get this you have to subtract your elevation using your depth to the well okay so now you get h you do this to another well so I walk to another groundwater well and I do another h I know the distance between the wells and then I would do the equation as q is equal to your minus hydraulic conductivity so since I am in the field I would ask what type of soil it is and then I know the hydraulic conductivity from the values I showed from Fries and Sherry book I know the let's not do the area of your well so let's do your q small q okay so we don't need the area of cross section we just need the discharge velocity so let's say q is equal to minus k times your dh which is your distance the difference between your hydraulic heads by the length the distance between your wells so all this we have done just by calculating two values which is your elevation of the well and then the depth to the well to get at your z which is the elevation of the well from the measuring point then we measured the groundwater level to get at three one minus two got me z and z plus three gives you h okay so with this we would calculate the groundwater hydraulic head for one well we do it for two wells and then we establish the equation Darcy's equation but remember we need the hydraulic conductivity which can be obtained from the previous values which are shown here so always have this you can have this slide I use this slide always to measure all the materials are there so to any system in the world you want and take a government report to understand what is that particular soil and then go to this slide and get the value and you can put it here by measuring the ground surface the elevation of the z but which you get by subtracting the groundwater well elevation or the depth of the well and you get q so please note that we will also look at some q values in the next lecture thank you