 As we have seen in the hydrologic cycle, the precipitation falling on the surface of the earth either runs off over land to the streams or some part of it infiltrates into the ground. The part which goes into the ground either gets transpired by the plants and goes back to the atmosphere or some part of it percolates deep down and contributes to the water already inside the earth. So, today we will look at this water which is existing below the ground surface. This is commonly called ground water or sub surface water. In surface water we have already looked at various kinds this may be pond, lake, streams or oceans. In the sub surface let us look at a cross section of the ground. So, let us say this is the ground surface and there may be a bed rock which is impermeable. So, water will be staying on top of this and this portion shown here by this line below this all the soil pores are filled with water. Therefore, this is known as saturated zone and if we take a soil sample from inside this portion the soil will consist of solid grains and the remaining portion will be all filled with water. In other words there is no air inside this zone while if you go above the ground water table or sometimes simply called water table. If we go above and take a sample from here it will have solid particles part of it will be filled with water and part with air. So, we may have a water flow around the grains and in the middle we may have air. So, this portion will be partially saturated and it is also called zone of aeration because of the presence of air. Now, in this zone again we can sub divide into three parts as we know that whenever water is in a tube it will have a capillary action because of which it will rise within the tube. This phenomena is known as the capillary rise and occurs because of the surface tension same thing happens below the ground level also. So, in this partially saturated zone there is a thin layer which is known as the capillary fringe. We know that the height of the capillary rise depends on the diameter of the tube. Therefore, the soil diameter will decide how much will be the capillary rise. So, this thickness of this capillary fringe will depend on the type of soil for smaller grain soil typically the rise is more for large grain soils the rise is smaller. If we go and look near the ground surface there would be a zone just below the ground surface from which the plants can take water and this is known as the soil water zone. The roots of the plants they can take water from this zone and this goes back into atmosphere through transpiration. This zone is important more for an agriculture engineer, but from water sources point of view we would be mostly concerned with the saturated zone or the ground water. If we look at the zone inside between the soil water zone and capillary fringe we can call it an intermediate zone. So, this is how water is distributed below the ground we have soil water or soil moisture then we have intermediate zone below that we have the capillary fringe and below that we have the saturated zone in which we are interested because this is the zone which will contribute which will store and contribute to the water when we want to use it. So, if we look at let us say this soil volume which has solid particles the size of these grains will depend on again what kind of soil it is. If it is sandy soil the grains will be larger if it is clay soil the grains will be smaller and then the surrounding area is filled completely with water. So, there are two things which we are interested in how much water is there and how much can be taken out and how fast it can be taken out. So, the questions which we need to answer is how much water is there this is the first question. Second is how much can be taken out because all the water which is present there may not be taken out because some of it will be attracted to the grains and will not be available. So, from a water sources engineer point of view we have to look at how much of this water is available for us and the third thing which is also important is how fast can this be taken out. Another thing which is important also is how deep is the water table because the deeper it is the more pumping cost we need to use and therefore, how deep is the water table. So, the depth of water table is also important. So, these questions need to be answered before we try to utilize this water source which is available within the ground. If we try to compare surface water and subsurface water there are some advantages in surface water some advantages in subsurface water. For example, if we want to use surface water from river typically it has to be carried to the area where it is to be used. So, we need to store the water somewhere then we need to carry it to the population which needs that water. So, we need a storage system and we need some conveyance system typically pipes networks for, but in ground water we really do not need the storage because everything is stored under ground the storage is already available to us. Similarly, for conveyance also we do not need structures because if we store water inside the ground it will diffuse because of the head difference and it will reach to the surrounding areas without laying any conveyance system and therefore, the cost of a storage and conveyance system is not there. So, it will be less expensive on the other hand we need to pump it because typically it will be available at a very great depth. So, if this is the ground surface the water table will be somewhere here and then there is a bedrock. So, if we need to take this water out we need to install pumps which will be expensive and then we need to constantly input energy to take this water out. If the water table is close to the ground level then we can just dig a well and get water from the dug well, but if it is very deep then we have to have some kind of bore wells which will be able to go very deep and we have to install a pump to pump this water out. So, let us look at these things one by one first how much water is there. In soil mechanics we have seen a term called porosity which is defined as the ratio of voids volume of voids total volume of the soil sample. So, if this is the soil sample and the total volume of this sample is V then we need to know what is the void space we call that volume as V V and the ratio of these two is known as the porosity. So, per unit volume how much pore space is available is the porosity and if it saturated with water then that means that much water is present typically we denoted by symbol eta. So, this much water is available it is not available, but it is present there how much is available that we have to see, but this much water is present in the soil sample of a unit volume. Now, all of this will not be available because some of the water which is very close to the grains is attached quite solidly to these particles and cannot be removed. So, in order to look at the availability we have a term which is known as a specific yield. Now, this specific yield which is denoted by s y is the amount of water which can be drained under gravity. So, if you take a soil sample saturated with water and then allow it to drain after draining there will be some water left inside the sample attached to the grains and may be sometimes between the grains, but the other water can be taken out and this is known as the specific yield of the formation. So, we can define it as the amount of water drained under gravity per unit volume of the formation. So, specific yield is the term which is more important to us than porosity two soils may have the same porosity, but they may have very different specific yields. For example, sand may have a porosity of let us say about 0.3 and a specific yield may be of the order of 0.1 or 0.2 while clay can have a high porosity sometimes it may be as high as 0.5, 0.6 or even higher, but the specific yield is typically small. So, may be 1 percent or 0.05, 5 percent. So, we can see that a clay formation although it can store a lot of water, but the amount of available water to us will be small and therefore, we would prefer to have a sandy formation in which there is storage also and it can yield a sufficient quantity of water also. So, a formation which stores water may not be able to yield water. The other term which we sometimes use although for us specific yield is more important, but specific retention is the amount of water retained and therefore, S y plus S r this is the specific retention S r. So, out of total porosity if we take out S y then the amount retained by the soil is S r and a specific retention typically will be higher for clay and less for sand. So, what we are interested in is the aquifer or the formation should be able to store enough water and it should be able to yield enough water. So, based on these two criteria we classify the formations in four different categories and these categories are aquifer which is the most important for us because it can store also water we would call it as porous. So, porous means that it has lot of pores and therefore, it can store lot of water and it can also transmit. So, once it is porous it can store water and it is also able to transmit water at a good rate. Now, this good term is qualitative it may change from place to place some times a class soil formation may be yielding a good rate for our purpose in some area, but the same formation may be yielding water at the same rate, but it may not be considered good for some other purpose. Therefore, this is qualitative and it will depend on the purpose for which we are using the water and porous we have seen already that this depends on porosity and specific yield. Porosity of course, is important, but specific yield is more important in water sources engineering because this is amount which is available to us. For transmissivity or how much how fast the formation is able to transmit water we have a term which is known as the permeability or hydraulic conductivity. We will look at this in details later, but right now we can just think about these as the ability of the formation to transmit water. So, if the hydraulic conductivity is larger that means the water can be transmitted at a faster rate for the same conditions. It has a porosity which is high or specific yield which is high and it has high hydraulic conductivity also. The example of an aquifer would be a sandy or gravel kind of formation. So, the aquifer is able to store and transmit water at a good rate. The second kind of formation which we classify is known as aquifuge which is just opposite of the aquifer. It cannot store water and it cannot transmit water also at a good rate. So, non porous and we can say the transmissivity is very small or impermeable. So, these are of course, not good for our purpose because they do not have enough water and they are impermeable also. So, we cannot take whatever little water is available. We cannot take it out very easily. So, examples of this aquifuge will be rock which has very small porosity and very small conductivity. So, this of course, is not useful for our purpose. Aquifer is the one which we will be using most often. So, this term aquifer we will be using to refer to all soil formations and which are of use to us. The other two classifications are aquiclude and aquitard. So, aquicludes are porous and less permeable. So, they can store water for example, clay, but the conductivity is very small and the aquitards as the name suggests retarding the velocity. So, it is a very low velocity formation. So, out of these as we discussed aquifer is the one which we will be targeting for our purpose. So, we have to answer the questions as to how much water can be taken out and how deep will we have to go for pumping the water. In other words, for any aquifer this is a ground level or a table and there is bedrock here. This is a ground level. So, we need to know if we put a pump here, how deep we have to install the pump because that will affect the cost of the well, deeper the well more will be the cost. We need to know when we pump water out, how much can we pump. Now, of course the pumping rate will depend on what is the transmissivity of the aquifer, but if we pump more the ground water level will go down. There will be some recharge to the aquifer which denotes how much is the deep percolation from whatever precipitation occurs in that area. So, that recharge if we want to be on the safe side then our pumping should not exceed the recharge, but typically it does not happen because typically we need the pumping in areas where the rainfall is low or in added zones and therefore, the recharge is very small. Therefore, typically the pumping rate is much higher than the recharge and slowly the ground water level starts to go down. So, with time we will see a lowering of the water table. So, we need to be able to answer the questions as to how much can be a safe value of q which will not result in excessive drawdown of the water table. So, in order to analyze this behavior of the aquifer we have as we discussed earlier we have this property of hydraulic conductivity for the aquifer which we must find out because this will govern the hydraulic conductivity typically denoted by k will govern the rate at which water can be taken out. So, aquifer description can be thought of in terms of its porosity, specific yield, hydraulic conductivity or permeability. So, these are some important parameters of the aquifer for an aquifer which is shown here in which the water table is open. That means the pressure at the water table will be atmospheric. So, if we look at this point or this point or this point the pressure will be atmospheric we can call it 0 pressure. Of course, above this there will be a capillary fringe which will have negative pressure, but we will not be concerned about this capillary fringe we will just assume that saturated zone is there and we will call it the water table and look at the behavior of water flow in this. Now, if the water table is open to the atmosphere we say it is an unconfined aquifer and the behavior of this unconfined aquifer is very different from the other kind of aquifer which we call confined. In confined aquifers if this is the ground level we may have a layer of impermeable material which will. So, this is the bedrock or there may be another impermeable layer below this, but this impermeable layer confines the water. So, this area is saturated, but because of the presence of this confining layer the aquifer water is under pressure. So, this water is under pressure compared to this which was open to atmosphere or it was not confined from the top therefore, the pressure will be atmospheric at this point or we can say. So, an unconfined aquifer on the top the water level will be subjected to zero pressure atmospheric pressure in confined the flow will occur under pressure and again depending on the pressure difference the water will move from one location to the other. And in terms of the movement of water we can define a head which is known as the piezometric head. So, let us look at this new term which we have defined a piezometric head which just means how much water will rise in a piezometer which is placed in the aquifer. So, if we look at let us say let us first look at unconfined aquifer ground level water table ground water table and bed rock. Now, if we put a tube of water here sorry a tube here the water will rise in this tube up to the ground water table because the pressure is atmospheric there, but if we take a confined aquifer a confining layer and then rock. If we put a tube in this water will typically rise above the confining layer because this water is under pressure and it may be coming from some area recharged at this location. So, this water is coming from let us say higher elevation. So, the head of the water will correspond to in a very simplistic way we can show that this water will rise up to the level from which it is being recharged. So, this is known as the piezometric head and it is the difference in the piezometric head which governs the flow of water. If there is no difference in the piezometric head then there will be no movement of water. So, if we take another tube here another piezometer here the water may rise up to this point. So, there will be a head difference which will cause the flow. So, if we plot the piezometric head it may look like this and the gradient of this typically denoted by I is called the hydraulic gradient. So, in unconfined aquifers the hydraulic gradient will be nothing, but the slope of the ground water table because if we put a piezometer here water will rise up to the ground water table. So, the slope of this ground water table will give the hydraulic gradient, but in confined aquifers the piezometric head difference delta h and. So, this gradient is what is the driving force for the ground water flow and in a very famous experiment which was connected almost 150 years ago by Darcy in which a tube filled with soil was subjected to some water flow of let us say amount q in units of meter cube per second or feet cube per second. And what the experiment looked at was if this is a tube cross section area is a the head difference between these two piezometer delta h and the length of l. Then what Darcy found that the amount of water flowing would be proportional to I where I is the hydraulic gradient and the area of cross section. So, if you take different experiments with different hydraulic gradients the amount of flow q would be directly proportional to the hydraulic gradient I and area of cross section A. And therefore, he introduced a constant of proportionality and called it permeability or hydraulic conductivity. So, the dimension of this k will be the same as velocity that is meter per second or feet per second. So, if you look at this equation we have a way of defining the hydraulic conductivity. For example, if you say that there is a unit cross section area tube subjected to a unit hydraulic gradient then q will be equal to k. So, hydraulic conductivity can be defined as the amount of flow passing through a unit cross sectional area of soil with a unit hydraulic gradient. So, we can write k equal to q for I equal to 1 and A equal to 1. And what it tells is that if hydraulic conductivity is higher then we will have more amount of flow from the soil for any given hydraulic gradient or given area. Now, if we look at the value of k we say that it has a velocity dimension the q is the amount of flow which is in terms of volume per unit time. So, if we divide q by the area we would get a term which will be similar to velocity. So, we can write this term q over A is an apparent velocity and why we call it apparent is because the whole cross section area of the tube suppose this is the cross section area A of the tube then this area consists of soil particles as well as the pore space filled with water. So, when we divide it by the area we are not accounting for the area of flow the area of flow that is the area from which the flow is taking place will not be equal to A, but it will be equal to the porosity times A. It can be shown that the volumetric porosity and the cross sectional area porosity they are same value. So, therefore, the area of flow actually the pore space or the amount which is occupied by water is smaller than A by an amount which is equal to the porosity. So, the real velocity which sometimes is also called seepage velocity will not be equal to q over A, but will be equal to q over eta A. Generally the term apparent velocity is used, but sometimes we also call it Darcy velocity and denoted by small q. The real velocity or the seepage velocity is also written as q over eta because q over A is a Darcy velocity. So, this is the actual velocity of flow and what it means is that if you take a soil sample and let us say we put some object here we inject a die then the velocity at which this will move will be the actual velocity and we then use the notation V to denote the actual velocity. So, if we put an object in the ground water its velocity of flow will be equal to q over eta and not equal to the Darcy velocity. The hydraulic conductivity is therefore, a very important parameter and there are various ways of determining the hydraulic conductivity for different soil formations. Typically sand will have or gravel will have a very high conductivity sand will have a little smaller and clay will be very small. So, in order to have formation which yields enough water and very fast we should have a high porosity specific yield and hydraulic conductivity. So, in order to compute the amount of flow through an aquifer Darcy's law is used as we have seen Darcy's law tells that discharge q will be proportional to the hydraulic gradient in the area and the constant of proportionality used is the hydraulic conductivity. So, if you look at let us say a confined aquifer. So, we have let us say these confining layers and if we look at the permeability of k for this and the area of A and let us say that these two piezometers are installed here which show a head difference of delta H in a length of let us also say that this thickness of the aquifer is B. Therefore, if we consider unit area we consider a two dimensional case in which we say that aquifer is extending for a very large distance perpendicular to this plane and consider unit width. So, unit width of aquifer then the area will be B into 1 we can write then k. So, this term k into B for confined aquifers occurs very commonly in these equations and therefore, it has been given a symbol of t and called transmissivity. For unconfined aquifers ground motor table ground level for unconfined aquifers the thickness of the aquifer H varies from point to point. So, if there is a gradient here then we can say that we will have some head here H 1 or some height H 1 and a smaller height here H 2 depending on what is the location of the bed rock if the bed rock level is assumed horizontal then if there is a gradient like this H 2 will typically be smaller than H 1, but in confined aquifers assuming that this thickness B is constant k B can be replaced by the transmissivity. So, in order to find out how much flow can occur through the aquifer for a given difference of head we use Darcy's law and then we use the Darcy's law to get the continuity equation which tells us a mass balance over a certain element. For example, if we take an element like this which has let us say length of delta x there is some flow coming in here from the left hand side we can call it q and there is some flow which is leaving which can be called based on the Taylor series expansion we can write it as q plus del q by del x into delta x. So, the continuity equation says that mass has to be conserved that means the net influence of this volume should be equal to the change of storage. So, there is some water is stored in this element and if we assume that the porosity is eta and consider unit width perpendicular to this plane then the volume of water present inside this sample or inside this elementary volume can be given by we take a unit width here let us say that the width is 1 and also 1 perpendicular this plane. So, eta delta x will be the volume of water stored within this element. Now, if this mass balance has to be satisfied it leads to the continuity equation which says that net inflow equal to change of storage. So, if we take this confined aquifer in which we are saying that there is no change in the height. So, what we are looking at is a constant height b and let us say certain length delta x. So, if we look at this volume since there is no change in the volume you can compare this case with unconfined aquifer in which case suppose you are pumping water from this aquifer with time the water level will go down or in case you have a well here it may go down as this, but the area or the volume is changing with time, but here in confined aquifers the area and therefore, the volume is not changing with time. Therefore, the mass can change with a change in the density and here the compressibility of water as well as the soil will be important as we know the soil is compressible. Therefore, when you have a confined aquifer which is being pumped as water is released from the aquifer the pressure goes down. So, this aquifer which was subjected to some pressure will now be subjected to a lower pressure and therefore, the density of water will change and the soil density will also change. So, the soil matrix compressibility and water compressibility both these will affect the release of water from this volume and therefore, when we write the continuity equation for confined aquifer case the compressibility of soil and water have to be taken into account. For unconfined case since it is atmospheric pressure and water can be treated as incompressible under atmospheric or small pressure conditions. Therefore, compressibility of water and soil does not enter into picture. Here the specific yield is more important specific yield will decide how much water will come out of an unconfined aquifer, but soil compressibility and water compressibility will affect what is the yield or what is the release of water from the confined aquifer. So, when we derive the continuity equation it is by nature different for confined and unconfined aquifers and we would look at the derivation of the continuity equation using Darcy's law and balancing the inflow and the rate of change of storage within the volume. So, if we write the Darcy's law let us put the area on this side this is the basic equation which we have to use in all our computations of continuity or mass balance. Now, what it says is that there is a velocity which is proportional to head loss because this I is proportional to piezometric head loss. Now, in laminar flow we have already seen in pipe flow cases that for laminar flow the head loss is proportional to velocity and for turbulent there is non-linear relationship where head loss is proportional to velocity to the power n typically n is around 1.7 1.75. So, if we look at Darcy's law what it tells us is that the flow has to be laminar in order that the Darcy's law is valid and velocity is proportional to delta h or the head loss is proportional to the first power of velocity we need to have some kind of laminar flow within the porous media. So, let us look at flow through a pore and for simplicity let us say that we have these kind of soil particles and there is a tube a porous tube through which what is flowing the diameter of this porous tube of course will depend on the diameter of the grains, but we can define as in pipe flow we can define the Reynolds number as velocity some length which typically will take as diameter and the kinematic viscosity this diameter in pipe flow is the diameter of the pipe. So, here since diameter of the pore will depend on the diameter of the grains we can put the grain diameter here which may be let us say d 10 or d 50 or d 90 of the soil velocity we can put as the Darcy velocity and this nu is the kinematic viscosity of water. So, when we look at Darcy's law its applicability will depend on the Reynolds number in pipe flow we know that the Reynolds number r e should be less than about 2000 for the flow to be laminar in porous media based on experiment has been found that r e should be this is for pipe flow circular pipe and for a porous media using a grain size as the characteristic length the diameter here d r e should be less than 1 for the Darcy's law to be applicable. So, the applicability of Darcy's law typically will be limited to a small velocities in general ground water flow has a very small velocity. So, generally r e is less than 1 and therefore Darcy's law will be applicable, but in some cases Darcy's law will not be applicable for example, if you are pumping water from a well the velocity far away from the well may be small, but as water comes closer to the well the velocity becomes very large. So, it may not be valid in some portion here depending on the pumping rate if the velocity is very high. So, for high pumping rates very close to the well Darcy's law may not be applicable. Similarly, for flow through gravels sometimes we may get velocities which are very high and Darcy's law may not be valid. So, when we apply Darcy's law in order to derive the continuity equation we should be aware of the limitation that the Reynolds number should be less than 1. So, we have to check the Reynolds number and ensure that it is less than 1 and then apply the Darcy's law. The second thing which we notice in Darcy's law is this term k hydraulic conductivity. So, if we write q equal to k i the term k it will depend on both the fluid and also the medium. For example, if you have a medium which is fixed for example, sand for different fluids the hydraulic conductivity will be different. So, suppose you have sand now k for oil and k for water will be different for the same material. Similarly, if you have water then k for sand and k for clay will be different. So, the hydraulic conductivity term includes both the fluid property and the medium property. We will see little later that we can partition this into two parts and intrinsic permeability k into a term which will only depend on the fluid. So, we will do this next time using Darcy's law. So, we will derive the continuity equation. So, in today's lecture we have seen what is ground water, what law the basic law Darcy's law which governs the flow through the ground water and what should be kept in mind while applying the Darcy's law. We have also seen different kinds of formations in terms of their water bearing capacity and water transmitting capacity. And in the next lecture we will derive the continuity equation for different kinds of aquifers.