Hello everyone, welcome to NPTEL course on groundwater hydrology and management. This is week 5 lecture 4. This week we are looking at the parameters to convert hydraulic head to different parameters that can be used in the groundwater equation. We also looked at how the groundwater aquifer can be divided into sub-components and given the merits of the parameters that are available. We chose the equations to model the groundwater flow. In the last lecture, which is lecture 3, we looked at the confined saturated region and we looked at how Darcy's law can be applied and only data needed was the hydraulic head difference between point A and point B or between two wells and then the distance between the wells followed by the hydraulic conductivity of the system. In today's lecture, we would look at the remaining components is the unconfined aquifer. In the unconfined aquifer, we saw that there is two regions. There is a zone of saturation which is under the imaginary water table line, which is this one and also there is a zone of aeration where the porous space has air plus water or just air and we understood that in the unconfined aquifer, which is the aquifer above the permeable layer, below the permeable it is called confined, above the permeable it is called unconfined aquifer. There can also be a saturated zone which is full of water, the pore space and no air present or negligible air and that part is characterized by an imaginary water table line. We also looked at that the Darcy's equation which was well accepted in the confined aquifer can also be used in the saturated zone of the unconfined aquifer. So we have seen section 2 and session 2b which is in the confined layer. So what is remaining? What is remaining is how water moves if the water table goes up which is the unsaturated zone and in the unsaturated zone still there is some water present and the water if connected between the pore spaces can move. Also we should understand that this water table line which is imaginary can be recharged and go up or can be discharged and go down. Let's say if it is recharged and goes up then what happens is this distance, this distance where you see or the thickness of the aquifer which was unsaturated becomes saturated. Here we have extra saturation because the water table has gone up. So there is no definite region in the unconfined aquifer. There's no definite region where we can clearly say above this or below this it is saturated. We need a model that can capture this dynamical movement of the water table. Basically we need an equation that accounts for the unsaturated zone also. Remember that Darcy's equation only holds in a saturated system. If you had the experimental setup by Darcy, Darcy initially saturated the soil and then applied a flow into the soil. That is why the applied volume which is Q is equal to the exit volume which is Q. So Q in is equal to Q out, mass is conserved which is one of the key conservation laws for Darcy's equation. Moving on, this cannot be true in real life especially in India where the fluctuation of the water table is there and we are no longer pumping only from the saturated fully saturated zone. We are also bringing in the unsaturated water. That can also be applied to the confined aquifers because our pumps are now into the unconfined aquifers which is this one and also into the confined aquifers. So the wells which are in the unconfined are called shallow wells or dug wells and the wells which go deep into the aquifers and into the confined aquifers are called deep bore wells or bore wells. They're not normally the wells that you see which is big and you can swim and stuff. It is like in the villages some wells you can swim and you can see the water levels. The deep or the confined aquifers you normally have a bore well which goes in deep as seen here. So once you start extracting the water, this aquifer is no longer fully saturated. There are regions where it becomes unsaturated also. So there is a need for a unsaturated groundwater equation. So let's look at it just for the explanation. We'll just look at it in this region which is in the unconfined aquifer which can be moved up and down depending on the water table. The water table if it recharges goes up. The water table if it is discharged or pumped goes down. So that is where your unsaturated zone boundary can be pushed up and down. Moving on, let's look at the equation which is going to be applied for the confined zone which is already sonamidazis. We're looking at the unconfined zone and especially the zone of unsaturation. We will also look at how the same equation can be used in the saturated. So what is the difference? The difference is the porosity is there and the water occupied is equal to the porosity 100% in the saturation whereas it is a function of water availability in the unsaturated zone. It is neither full 100% nor 0%. It is in between somewhere and it fluctuates depending on the water taken up by the plants pumping and also recharged. So when recharged happens more saturation happens and water is pulled out there is more discharge happening. Let's look at it. We're going to very focussely look at transient unsaturated flow. As I mentioned the steady state is only when you have a fully saturated zone and also water comes in and goes out because you're maintaining the head differences in both locations. Suppose you don't maintain the head it becomes a transient situation and the transient situation is normally occurring in the unsaturated zone because the saturation is different and so the water volume may not be the same across time the flow magnitude and the flow velocity. It may not be the same across time because there is a time when water gets absorbed in the soil and there is a time when water connects with the other soil pores and then starts to move. So it is not constant in time that is what transient is and in unsaturated zone it happens naturally and why we're looking at this in detail is because most real life scenarios are unsaturated because we pump and then recharge and then pump and then transient in nature because the head is not maintained because you constantly disturb the system. So whenever there is disturbance it is transient. When it is naturally flowing like in a forest in a dammed area where the head is always maintained and always you have the same pressure that the system has stabilized then it is steady state where there's not much disturbance happening but since the core of this course is going to be looking at the disturbances in the groundwater aquifer and how to manage this is going to be a very very important situation. Moving on we have to define what happens in the unsaturated medium. The degree of saturation is not 100% which means the pore space inside the pore space there could be some air volume present and or water volume present so it is not 100%. When it becomes 100% it becomes the pore is not only a function of volume but also the degree of saturation. So if the soil moisture is fully saturated then you have a Darcy's equation that can apply because whatever water comes in can be pushed can push the volume inside the soil and another volume can come out. However if the soil moisture is not fully 100% the saturation is not 100% then some of the water would first align itself and start filling the pores. Some of the water would be used for wetting the soil we call it. So in that case you would be losing on some flow because you have to compensate for saturation. So the flow is not only a function of volume and time which is t0 to t1 and also the volume applied in the top if you see Darcy's setup you have q coming in and that q drives the outflow also but here it is not only the q but also the inside tube soil material saturation is an important factor. What do you mean degree of saturation? As I said zero means zero saturation means it's totally dry and saturated fully unsaturated and then if it is 100% it's called saturated but normal real life scenarios it is normally in between both and we call it partially saturated and that is a zone of aeration etc. So this is absent in the Darcy's approach we won't call it a limitation because Darcy initially did this equation for a pipe supply for fountains okay and in the pipe and wherever the pipe was laid it was fully saturated because they had enough water it was unlimited water which was supplied so that the fountains work you have the fountains coming. That is not the system in the groundwater right so it was an accidental discovery Darcy did for groundwater equation however we it has been widely used and it has been successful in dealing with the saturated flow both in the unconfined and confined so right now we have to take a note of it and say thank you but there are some issues in applying it for unsaturated flow let's look at it. As I said there is nowhere in the equation q is equals to minus k del h by del l there is nowhere the saturation of the soil comes into picture you have the hydraulic conductivity which gives you the ease of which water can flow through the porous medium but there's nothing related to the actual saturation of the soil okay so writing it down here so q was equal to your minus k and del h right so here there is nowhere you have a saturation attached because your k was only a function of how easily the material allows the water to flow or the fluid to flow so it is a function of the pore space and also the fluid and here it is water therefore there was another equation which was built it was also built drawing some influences from Darcy it was made by the soil scientist Richard's and because he did so much work on it the equation is named after him as Richard's equation Richard's equation has a another parameter called psi which is a function of the porous space saturation it is introduced into the previous equations derived from Darcy so to understand how the degree of saturation can play a role in your groundwater equation it is more difficult just look at the equation how it has been spread out the definition of the equation would not be discussed here but if you could look at it initially initially the k was only a function of x y and z which is the plane but here the k is also a function because here you can see that it is within the x domain this is the y domain this is the z domain so this is clearly kx x k y y k z z but you also have a psi function the hydraulic head is also introduced into it so the psi here gives you the degree of saturation and also it is a function of the saturation in the porous media in other words you could look at it flow as if the saturation is high there will flow there will flow up there suppose the saturation is zero psi goes to zero and all this collapses you're saying that q is zero so this is the basic setup of groundwater models wherein you have some complex models which uses Richard's equation and some very basic models which is using only the Darcy's equation there are takers for both these equations because computationally we'll see how difficult it is to use Richard's equation because you need to give the data how psi varies with x you need to know and also how psi varies with time also we need to know which means there's a lot of data in the Darcy's equation there's only one variation which is the hydraulic head how the hydraulic head between A and B varies with time is all you need to understand the flow here not only the hydraulic head but you should also know the in-between soil media and how the saturation is present for example you have a well there's water inside then you have another well there's another water inside however if there is no saturation between that how will water flow from A to B it won't right because most of the water will be used to saturate the system and then flow so that degree of saturation is captured in the Richard's equation however Darcy requires that there is a medium present between A and B the wells A and B which is fully saturated moving on most studies still use Darcy's due to simplicity so in the real-life scenario where you are using models please understand that you can make it as complex as you want you can throw in more numbers you can throw in more equations however the computational difficulty goes up the requirement of data goes up and sometimes the model crashes because you don't have the data in other times if you don't have enough data the model will try to assume a lot or you force the model to fit the hydraulic head which is also wrong so most of the real-life studies where you have water for groundwater recharge and other things or check dams and monsoon recharge rainfall recharge or summer extraction if you see that most of these studies still use Darcy and if you know Darcy's was done in the 18th century very very old and still the law is used because of its simplicity and yes it has some issues but overall computationally and data intensive it is less and it captures the groundwater flow to a particular extent. So as I said in Darcy there is no function for Psi where it is a degree of saturation whereas here you do see that Psi is present throughout the equation and even your hydraulic conductivity is a function of Psi because you can have high hydraulic conductivity but if there's no saturation there then how will water flow is raised by reaches all our valid points however the degree and change of Psi of came with Psi and or how it varies across time you can see d Psi by dt is does take a lot of data. So in that note and as I said that there are many studies which still use and prefer Darcy's law let's look at Darcy's law's strength okay Darcy's law provides very accurate description of the groundwater flow in almost all hydrogeological environments why because the only data that it requires other than the groundwater data is k which is your hydraulic conductivity and as I've already shown in class if you don't for example I don't have to go to the field to find k what I do is I quickly understand what is the geology present what is the type of matrix which is a soil or the type of geology present and then I go to my book which is the groundwater book by Fries and Cherry where all the k values for any material in the planet is given there's no these these rocks and soil do not emerge which means they don't evolve it is the same thing only evolution is it weathers it weathers from the parent rock into soil so all this has been documented very clearly in Fries and Cherry's groundwater book and that data is still used for assuming or or estimating the hydraulic conductivity and even though there is a range you always get away with having the average or the middle of the range for most of the materials so Darcy's law is simple less data intensive and all the accompanying data is already well studied in the literature and backed up with data so Darcy's law has provided an accurate description in all hydrogeological settings where it is mostly saturated where does it law hold good Darcy's law hold good in saturated flow as any state flow where the hydraulic heads are kept constant so that there's always continuous movement of water and magnitude and well at a single point doesn't change over time it also law holds good in the transient flow where you change the hydraulic head and the transient there is a transient flow occurring wherein the flow magnitude and velocity changes over time please understand that it changes over time because of the changing hydraulic head conditions and pumping and recharge regimes not because of unsaturated here in this in this example I'm saying so in transient flow also Darcy's law hold good as long as the medium is saturated so saturation is the key it has to be saturated and then there is steady flow and then there is transient flow okay for flow in aquifers and for flow in aquitards so aquifers as I said it can hold good in unconfined aquifers it can hold good in confined aquifers because both have saturated conditions and it can also hold good in the impervious layer which is the aquitards please understand that recharge also has to go through these aquitards to get into the confined layer it is very small very small amount of water that goes in however Darcy's law can still hold good in that because hydraulic conductivity you know you know the hydraulic head up and below so for example if water moves from here to down you know the water level at A and water level at B water flows from high potential to low potential the material is the aquitard which has a particular hydraulic conductivity then check it on the freezing cherries groundwater book and you can estimate groundwater flow it holds good in Darcy's law holds good in flow in homogeneous system and both heterogeneous system also homogeneous wherein the material is homogeneous so that the flow conditions are homogeneous across and it also holds good in heterogeneous systems where you have different types of materials thereby affecting the groundwater flow rates and these can be captured by your changes in hydraulic conductivity a homogeneous system let's say it's a sandy aquifer across across the two regions A to B there is sand and I have one value of k it is fine so you can estimate Darcy's law groundwater flow using Darcy's law suppose the material is heterogeneous which means I have sand and then clay and then sand still Darcy's law holds good because you can have different k values so it will be first compartment and then the clay compartment and then another sandy compartment so you can just add them all together to get the flow in the heterogeneous system it can also do flow in isotropic media and flow in ananisotropic media please remember what is the difference in isotropic kx is equals to ky is equals to kz it doesn't change however in an anisotropic way in the medium the kx ky and kz need not be the same and the changes between two points need not be the same so in such a system you can still hold Darcy's law good because all you have to do is dissect the equation one equation q is equals to minus k del h into three equations basically because you have kx ky and kz we have seen this in the matrix kind of solution for Darcy's law it also holds good in both rocks and granular media what is rocks rocks are the deep deep aquifers where you have the porous space still present but it is mostly fractures and also less connected porosity so that is the rock medium the granular media is about the rock where the rock has weathered and soil has formed so both in the granular media and the rock media the Darcy law holds good this is also a example of a homogenous and heterogeneous system a homogenous is purely rock for example in a heterogeneous system you can have rocks and and your granular media also so Darcy's law has well proved to be a very successful groundwater flow equation across centuries now remember it was done in 1800s and it has been used till date even I use it till now the Darcy's law even for lab experiment and for your real life field experiments Darcy's law holds well the only very very important assumption or condition is saturation it has to be saturated still some people have used it for saturated systems for example mod flow is a very well developed model initially it was only doing Darcy's equation even for unsaturated flow it just turns off and on some layers however it has given good values so this is where you can push the science to higher limits however at one point it has to stop because you just keep on adding equations it doesn't improve the efficiency of the model higher it just makes it more cumbersome in terms of media so we've seen all the assumptions and we've seen all the strengths of Darcy in the next class I will also go improve the limitations of Darcy's law and the wrap up of week five so in this week we have seen two equations for groundwater flow estimation one is Darcy's and the other one is the Richards both can be used in saturated system Richards can also be used in saturated system but where Richards becomes more beneficial is it can model the unsaturated zone well Richards equation is also old it's not a very new equation so it has taken time it has proved itself to be a very good equation and if you have the data and computing power Richards equation is the best in terms of accuracy for simplicity Darcy's law is the best and worldwide acceptance studies how many studies have been done Darcy's law holds good even till date I will see you in the next class on the limitations and future directions for promote equations this I would stop just showing what is going to come for the next lecture basically the representation of Darcy's in space I will see you in the next class thank you