 Welcome to groundwater hydrology and management. This is week five, lecture five of the NPTEL series. In this week, we have been looking at the groundwater equations in two different systems for aquifers, which is confined and confined. We also defined some parameters which were useful for assessing the groundwater equations. In the last class, we looked at Darcy's application in the strength of Darcy's law, which is mostly for saturated system. We'll continue with the discussion on what are the limitations. Before that, let us define the volume in which Darcy's law holds good. What are the assumptions? Darcy's continuum and representative elementary volume is a very important aspect to consider while using Darcy's law. What does it require? It requires a replacement of actual ensemble of grains that make up porous medium by a representative continuum, which means instead of having particles and spaces inside the particles, it is being replaced by a continuous medium. It is called Darcy's continuum. It is not a porous space. It's not like a sponge with different holes and then water flows through it. It is a continuum volume where water can flow through the volume and come out in a more linear fashion. It's like driving through a tunnel. Again, please understand that this was made by Darcy to understand the flow through pipes and fountains, so his assumptions were valid. However, they have been also valid for groundwater systems at a particular representative elementary volume, ReV. So we will go to define this ReV, which means at what volume does Darcy's continuum approach be valid. We also would like to understand that the reality is we have solid grains and inside that there is porous spaces. So water would move not in a continuum, but it goes down up and then sometimes gets connected. Sometimes it doesn't. In a Darcy's volume, it is a continuous volume where connections are always made and that is why the saturated condition is made during Darcy's law, which means the medium has to be saturated. Once the medium is fully saturated, then the continuum can easily take place because eventually the water will be connected and it will flow through the pipe or in our case, it is a solid medium. The other very important assumption which is needed for Darcy to work along with the Darcy's continuum and representative elementary volume is the macroscopic view rather than the microscopic scale. So if you look at it, once you have a small very, very microscopic scale of analysis of the soil, you see the porous spaces and the solid grains. But once you go to a macroscopic view, you don't see the individual porous spaces, but you see a continuum and that is all these assumptions are intertwined into one, which is the macroscopic view. Let's have a look at it. We've already defined this in the earlier classes. So in a normal soil, water particle would move up and down in a tortuous path. It will flow through the least resistant path. It will find where the least resistant path is and normally it is not a straight line. It has to go around and it takes time and then comes out. This is the microscopic view. However, in a macroscopic view, I don't care about the individual porous. I don't care about the individual solid particles and their shape and how they are arranged. However, I take a volume and inside the volume water goes and comes out. That is the macroscopic view. And all I need is the cross section of the macroscopic area and an average hydraulic conductivity to estimate Darcy's law, which is q goes in. What is the q that comes out? So this microscopic versus macroscopic view is very, very important to define early in your groundwater equation. And that actually makes you pick and choose which model you're going to use. So in a Darcy, it is mostly macroscopic. And once you do the macroscopic also, you need to define the element of volume of analysis, which is this one. Because if it is too small, then other forces also contribute to throw Darcy out. We will get into those aspects now. The other thing which is also important in the microscopic versus macroscopic is the actual flow path of the particle as you see is through different channels rather than one single continuous channel. The water can go up and then come down and then go into branches and then come out eventually. However, Darcy's approach assumes a average linear flow path, which means the actual flow path can be very torturous. It goes up and down, loses velocity, gains velocity. However, the average of it is taken as a linear velocity in Darcy. The other aspect in Darcy is a uniform porosity, which is assumed in Darcy, which means you have a solid material, let's say a block. And if you cut slices along the block of the soil material, you have one slice taken out, which is NA1 and NA1 is not the same porosity as NA2 because it could be compaction. There could be some other roots growing. So for example, a plant is growing. The soil underneath the plant will have more pore spaces because there is roots which go in. However, if you take a sample outside of the plant area, the root zone, your porous space would be less. So in reality, in real world, the pore space is not the same across a sample. It differs. So how do you assume that it is the same is what the question against Darcy is. But as I said in the previous lecture, you can keep on probing these inaccuracies and assumptions. But however, we have to stop at one point. Let's say I take a soil. I have plants. Okay, this is the top surface. I have plants here. And after this side, I have no plants. How many samples can I take to assess the porosity? It's time consuming. And by the time you do it, your groundwater flow equation is no longer needed. And there's a lot of soil disturbance also. When you take the soil, you're actually disturbing the soil. So it is very important to understand that macroscopic view is justified. However, the justification comes with the accurate assumption of the soil particles. Let's take an example. I have a silty loam soil in one area for a cell growing. In one area, I have barren's silty loam. And then the other area, the silty loam has construction on top of it. And in the fourth, I have silty loam on rice. So the trees would have high roots, big, big roots going in. So you have high porosity. Then you have the rice field where it has the roots of the paddy and the rice plants. And so you have good porosity so water can move. In the next version, where I have the barren land, the porosity is the natural porosity. You don't have trees, you don't have plants. There is some porosity, which is intrinsic property of the soil. Whereas if you go to the constructed land, there is compaction on top of it. So the pore spaces are brought the lowest. So here's where the, as I said, the assumption comes in. So you've already seen a range in Friesen-Cherry's groundwater book. So you need to take the lowest, the lowest range for the salty silty loam for your area. Otherwise, if you take the highest, it will be in a wrong assumption. So what I'm trying to give you is there are uncertainties. There are issues when you do a macroscopic view, but you could justify it using correct assumptions from literature. Otherwise, you cannot defend it by going on taking samples and samples to assess a porosity. Let's take this example. The porosity of this is much, can be much smaller or bigger of the porosity of this layer. It's the same soil. I just cut it into two. Why should both of them be the same? This is a question, because if, for example, if there is a trees going on this side, on this first slice, which is Na2, the porosity would be much higher than no trees on Na1. This is where Darcy was very simple. He said, no, I know the soil. I look at the soil and I see no trees, no plants, no constructions. So there is a middle range of soil porosity, I will assume, and that assumption, he just kept on checking with the model and it worked. Same thing here. I do not know how many, how many parts are active. How do you know how many parts are active? It's very, very difficult to estimate. And as the name says, it is microscopic, which means you have to look in a microscope to find all these flow parts. People use CT scans, which means they take the soil sample into a scan and it slices the soil to see these flow parts. It's important for certain applications, but for normal groundwater flow, you don't have to do it. You can get away with this average, but the average should be a very calculated assumption. That is what Darcy is good about. Moving on. So as I said, there is a Darcy continuum where it is not regularly connected flow paths. There is a continuous flow path and it is a macroscopic view. For the macroscopic view, there is a volume for each sample beyond which it becomes macroscopic. So if you look at it, you have the porosity on one axis and you have the volume on the x-axis. So when you have the smallest of the smallest volume of a sample, the porosity fluctuates a lot. This is the microscopic view. And when the porosity fluctuates a lot, you have these preferential flow paths. You have the actual velocities, flow paths in post spaces, which you see here. And you have changes in the aerial porosity. That is in the microscopic region. As you will see, the same material allows the post space can fluctuate between the volumes, between the samples inside the volume. Slowly when you increase the volume, the noise comes down because you're averaging it out. The noise is being averaged. So V3 is that critical volume threshold above which the systems become macroscopic, which means all this undulations and noisiness in the porosity between the samples are now able to be averaged. And that is the volume which makes the sample a macroscopic view. And it depends on the soil type, the type of use of the soil, land use, land cover of the soil, for example, plants, trees, or etc. And how it is managed. Is it built? Is it cloud? Is it just barren land? Those kind of things. So once you go on increasing the volume, it is still macroscopic. This is the same for a homogeneous system. In a homogeneous system, your porosity fluctuates very less. Whereas in a heterogeneous system, it fluctuates a lot. And there is a volume above which, again, the porosity is very important, which means it starts to fluctuate. Think about big, big rocks. You cannot have porous space between the rocks more than the ones you put along the beach to stop the water from going in and reduce the waves energy. So those cannot be taken as a big macroscopic volume. And then you can model the groundwater. You have to have a limit also to the macroscopic. So this is the volume. This is the key volume that is very, very important. Anywhere from V3 to V4, V4.5. After V5, the heterogeneity in the porosity would continue. Whereas the homogeneous system is always homogeneous. It is always the same. So if you look at Darcy's equation, it assumes homogeneity. Soil particles are well arranged, are saturated, and homogeneous, which means the porosity will be the same across the system. Whereas in reality, it is not the same. For example, an earthworm might have gone into a soil and that gives a flow path which is suitable for groundwater to flow. The idea is when you go in a very small scale, these issues would go up and your groundwater flow model would crash. Your accuracy is lost. However, when you take this volume, a high, high volume for a groundwater sample, then your variations are all averaged out because this average would become down. So this and this would can come around here and then goes down and here it is almost less. So this is the porosity, I can say, is the accurate porosity, which is the average of the noise after a particular volume. So this volume is very important, which is called the representative elementary volume. In a microscopic, the volumes are very small, and that is why it's called micro. In a macro, which is big, the volume is big, and all the porosity noises are averaged out. So with this, even though Darcy's law has been widely used, there are some concerns and limitations. Let me go through them. So to teach you all these basics of science and models and equations, please understand that you need to share also the limitations whenever you're doing this kind of work. Because if you say that now I'm going to model nature accurately, then it is wrong because nature is very complex. And you're using a mathematical equation here. It can be a physically driven equation like the Darcy's law, or it can be an empirical equation like your E.T. calculations. Either way, you have to give the limitations so that it shows that you have good understanding of the equation. So what is the limitations in Darcy's law? It is microscopic view. So in real life scenarios, it is troublesome because groundwater flow occurs also at the microscopic level and at the microscopic level. So the microscopic level assumptions are made for Darcy's law, which is a limitation. So it is a macroscopic view, is a limitation, saturated versus unsaturated. The assumption of fully saturated in the medium is always held tight in Darcy's law. Any experimental setup for Darcy you see would have the soil in medium or the rock or whatever the porous space fully saturated with water. And then the equation is applied. As I said, it can be good for a pipe network. It can be good for fountains and lab in lab assessments. And it has been good in the real life scenarios. But the reality is that it is not always fully saturated. So as I clearly as mentioned here, soil moisture is not always 100%. And it is not always steady state. Steady state can be in a system where the disturbances are low. Once you disturb the system, the groundwater will fluctuate and then come to a new steady state. Then you disturb the system, it fluctuates like recharge, pumping, etc. And then it comes back to a new steady state. The steady state can be the same or a new equilibrium, new steady state reached. And we know that groundwater has been pumped so much across the world. So the steady state won't be reached. It is a transient in nature. Also groundwater flow can be turbulent in nature. It cannot be very smooth and going in a particular velocity. It can have multiple velocities depending on the. Turbulent flow causes Darcy's law to crash. I will show you in the next slide how Darcy's law does not hold good during turbulent flow. Turbulent is where the vertex of the water vectors would not only go in the same direction, but it can go in circles, eddies, depending on the interaction with the soil media. In Darcy's, it is actually flowing through a pipe. It is an average velocity. It is a linear velocity. It assumes porosity is averaged out. So what happens is it is like flowing through a pipe. However, there is interaction of the soil particles or the rock particles on the water, which will cause the turbulence. If you go slow, turbulence is less. If you go fast, the turbulence is high. It is like biking through the air or let us say you are boating on a river. If you go very slow, you do not see the waves behind you forming any patterns. But if you go very fast, then you leave turbulence. It is the same like how planes go. Bus don't create much turbulence. But if you see a plane, it will go and then turbulence is formed. So that is a function of the velocity. Darcy's law is assumption is linear. Linear movement and linear velocity. This has big issues with in real life scenarios where the low permeability sediments are present under low gradients. So for example, you have a very slowly allowing water medium, low permeability solid. And then if there is already a low gradient, low gradient pulling the water, then the linearity won't hurt. Darcy's law is very linear. It is a function of your hydraulic humidity and hydraulic gradient. Assumptions is linear and it doesn't hold good when the low permeability sediments and low gradients are present. Because water can not only keep on moving it to stop, the low gradient doesn't pull enough. There is not much activity. So water can go in a lateral position or it can just stay there for some time. The same applies for larger velocities also. So either if it is too slow, the linearity is gone. If it is too fast, the linearity is gone. So that is the second assumption here. Issues with large flow through high permeability sediments. Like for example, the rocks I said, when you pour the water on the rocks under the ground, water just gushes through and it picks its own path. It's not a linear path. It picks its own path. It gets stored in one place. And if the high permeability is there, high specific yield is there, just water just gushes out. So therefore there are limits. So you should understand when you use the senior system and when you report these as a value using Darcy's law, you need to understand these limitations. For example, if you're applying water in a field, that is not a turbulent flow. It is a non-laminar flow and the sediments are not low permeability or too high permeability. So your Darcy law is good. But as I said, if you're pushing a big dam, can I just break the canal and setting the water into the field, then the equation won't help. But here we have specific discharge on the y-axis and we have your hydraulic gradient, which is dh by dl on the x-axis. So as the specific discharge and hydraulic gradient go up, which is slowly the velocity is going up and the hydraulic gradient, which is pulling the water is also going up. So Darcy's law is good. It is linear and laminar flow, slow well movement of water. But then there is a non-linearity and a laminar flow, but also there is a turbulent flow, which is given by the Reynolds number. So Reynolds number is given to understand the turbulence of the fluid and for water it picks up around 10 power 2. So once you have this turbulence, which is a function of these fluid material, the viscosity and also the surface tension, so you could see that how this discharge at high discharges and at high hydraulic gradient, your turbulence kicks in and that is where your Darcy's law fails. So at this region, your Darcy's law fails. Once the flow at the discharge becomes non-laminar, which is turbulent and non-linear, the Darcy's law application should stop. So you should not be promoting Darcy's law in high turbulent groundwater flows. But you don't get that, so that is the reality is you don't get it in much groundwater systems. Only in cast geology where you have limestone and inside that the water flows, that is very turbulent. You would have seen caves and underneath the water flows. If you take a macroscopic view, that is also groundwater, because it is under the ground and water just takes away the sediments and the solid materials, it dissolves the solid material, which is limestone and flows through it. If you apply groundwater flow there, it will not work. At that point, you should treat the flow as a surface water discharge and you apply the Manning's coefficient and other hydrology parameters to it. You treat it like a river, even though it is under the ground. So that turbulence is very, very important and it happens only in very, very specific terms. So in our class where we are looking at groundwater for India and adipose will use Darcy's law and this range still holds good. The only place where it doesn't hold good is the unsaturated situation and in transient flow situations. So to wrap up, the groundwater flow equations have been discussed in class. We saw two equations Darcy's equation and Richard's equation for which we have first divided the study area into a unconfined zone of aeration and unsaturated aquifer and then we had a saturated aquifer in an unconfined aquifer. So this is an unconfined zone, unconfined aquifer with an unsaturated setting and we had a confined aquifer with a saturated setting. So we also defined zone of aeration and zone of saturation. So to club it, what did we do? We said the Darcy's can be used on the top for unconfined aquifers for saturated and Richard's equation is best suited for the unsaturated condition where Richard's equation brings in hydraulic conductivity as a function of the degree of saturation. Whereas Darcy assumes 100% saturation. So the idea is when you have 100% saturation, your Richard's equation boils down to Darcy's. So that happens in the unconfined unsaturated area. So the unsaturated can also be saturated because when water flows and then recharges, it becomes unsaturated. So that zone, the zone of aeration, if it is saturated, please use Darcy. If it is unsaturated, you can use Richard's. Then we come to the unconfined region under the water table. Darcy is mostly used for both the conditions unsaturated and saturated because it is always saturated. At some instances, the water level comes down, becomes unsaturated and then picks up again. So Darcy's law has been used widely for that. In the confined aquifer also Darcy's law holds mostly good because Darcy's equation can capture both the unsaturated and saturated within the confined, assuming that some layers, if are unsaturated, you don't model those layers. You only model the layers or use the equation for the layers where the Darcy law holds good. Darcy's mostly applied than Richard's equation. There is no issues with that. You just have to be careful in telling that, yes, I use Darcy. Yes, I have these assumptions, but still I would get the results from which I couldn't understand what is going on. So to recap the week five, we looked at contour lines or equipotential lines which are basically constructed from the hydraulic heads. You place them across the field and then from the field you have hydraulic heads placed across the field. You connect these hydraulic heads of same levels to get contour lines estimated from hydraulic heads and gives an understanding of the gradient, the direction in which no water flow is occurring. And once you have the contour lines, and the gradient, you can next estimate the flow lines which are imaginary lines that traces the path of a particle of groundwater, how it flows from one point to the other. Then we saw some rules for homogeneous systems and heterogeneous systems. Then we also looked at groundwater equations and mostly groundwater and saturated confined can be given by Darcy's equation and groundwater in unsaturated and transient is given by Richard's equation. On the whole, we saw that Darcy's law can be used for most of the systems and even in systems where Richard's equation can be used. For example, as I said, before Richard's Darcy's law was used for everything. Richard's just added more efficiency, more accuracy. But before that, it was only Darcy's law and 1800s to 1900s. So Richard's came only 1900s. So across the years, still Darcy's law was used and well documented. And all the results are still okay. So Darcy's law is widely used, many studies done. However, there are a lot of limitations and strengths, which you should understand before deploying it for your study here. Richard's equation, the fundamentals are very good and it has a function of soil moisture, the degree of saturation, which is not present in Darcy. However, the data requirement is less in Darcy's law compared to Richard's law. So for that Darcy's law has been used widely across the world. With this, I would like to conclude the fifth week of lecture on ground water equations and setting up these parameters for the equation. I will see you in week six. Thank you.