 Welcome to module 2 of advanced geotechnical engineering course. In this module we are going to introduce ourselves and discuss in depth about permeability and seepage. So this lecture is titled as a permeability and seepage 1. As we have introduced already in the introductory lecture, the module 2 which is nothing but permeability and seepage, the contents are the permeability, seepage forces and effective stresses due to seepage, measurement of the permeability in the laboratory as well as in the field, Laplace equations of fluid flow for one dimensional cases and two dimensional and three dimensional seepage, flow nets, anisotropic and non-homogeneous medium and confined and unconfined seepage. Along with this we are going to discuss about some practical problems particularly with a tendam construction and also some canal embankments construction. Field flow through soils, the main motivation is that the ability of engineers to understand and predict the flow of fluids usually water. In case if the fluid is other than water that can be a contaminant then lot of work is actually happening in the unsaturated soil mechanics. So the ability of the engineers to understand and predict the flow of fluids usually water in soils is essential for many applications in civil engineering. Many geotechnical structures or hydraulic structures which involve the water flow through the soils. So the motivation is that the engineer has to understand and predict the flow of fluids in soils and which actually has got lot of prominence in many civil engineering structures. For example in the environmental engineering a holding lagoon we are interested if you are storing a toxic liquid in the holding lagoon we are interested how efficiently we can confine this toxic liquid in the lagoon. So if at all if there is a leakage happens at what rate is this toxic liquid escaping the holding lagoon and how long might it take the liquid to reach the groundwater table and what can be done to slow down the rate of escape of the pollutant which is nothing but a toxic liquid. So in this application problem we are interested at what rate this toxic permeant escapes through the holding lagoon or how long might it take the liquid to reach to the groundwater table and what can be done to slow down the rate of escape of the pollutant. In the other side on the other hand on the construction engineering particularly if you are constructing within rivers or middle of a flowing river or a flowing water body we are interested in confining the particular localized area which is done basically by cofferdams and the foundation for the either bridges or the structure which is being constructed will be made. So the temporary sheet file walls which can be rectangular in plant dimensions or can be in circular in shape or square in shape and they are actually constructed and then prevented the water flow into this area and that the area is actually available for construction of placement of the foundation. For example in this slide a typical cross section of a cofferdam is shown here and basic idea is that when we retain the water surrounding water and into the then what how you know the flow can actually take place is actually shown here. If you see here this particular portion the water enters the soil and then it actually flows up. So if there is some stresses which are actually high get dropped here and there is a danger of the foundation which is going to be constructed. In this particular slide a typical example of a cofferdam section is shown. So wherein this is in a river bed where we have actually got stratified soil. So the river bed starts at minus 0.81 meters here and there is one type of soil which is actually here clay which is fine grained in nature. So this extent is about 7 meters or so. Then we have got sand briefly which is named as sand 1 a type of sand about 3 meters. Then for about 7 meters we have here another sand type and then there is a clay. So if you are actually in order to construct a pier or a pier foundation what it is intended is to construct a cofferdam which is generally a sheet pile wall which is driven into the ground and thereafter after putting the necessary ties the excavation will be dumbed up to a certain level. And here in this particular case if you see where there is a river bed and above that there is a filling which actually has happened and then this is the high tide level or upstream water level what it is called. Then in that situation in this practical situation this structure stability has to be seen for the water flow from the high tide level to the downstream water level. So this type of practical situations are actually coming up now because of the ongoing infrastructure work which is happening currently in many parts of the world. So in the construction engineering important questions that would be that would need to be answered are what will be the rate of water flow into the site. So what will be the rate of water inflow into the to the site and is it possible that the soil will liquefy or lose the stress and endanger the construction workers. So our objective is to gain an understanding of the mechanics of the fluid flow in soils so that engineering problems of this type can eventually be addressed. So if you consider the fluid flow through the soils as it is actually shown in the slide water flows along the edges of these soil solids. So this particular path is actually the flow takes place along the edges and these are the voids within the soil solids. So water flows so this is the path subtended by the flowing water because of the availability of the some energy. So you can see here the flow which is actually subtended along the edges of the soil solids. So the soils are the basically assemblies of solid particles and with interconnected voids through which the water can flow from a point of higher energy to low energy. So soils are actually assemblies of solid particles with interconnected voids through which the water can flow from a point of higher energy to lower energy. So this study of the flow of water through porous soil media is important very much in geotechnical engineering because it involves the rate at which the water flow through the soil. So it enables us to determine the rate of leakage through an attack or involving the rate of settlement of a foundation. For example if there is a foundation which is actually resting on certain type of soil and if the water is escaping out of the soil it will tell us that how much time the settlements will take place or the rate of settlements of a foundation. And involving the strength that is the evolution of the factor of safety of an embankment. So depending upon the rate of flow of the water out of the soil the soil actually gains a strength. So it involves that the evolution of the factor of safety of an embankment. So in the construction from the fluid flow through point of view we are interested involving the rate at which the water flows through the soil that is to estimate the determination of the rate of leakage through an earth dam. And involving the rate of settlement of a foundation or involving the strength gain that is the evolution of the factor of safety of an embankment or a dam which is being constructed on a soil. If you look into this the water does not flow from point A to point B. So in this particular slide these are the soil solid particles and point A and point B which is actually shown here and this is actually called as the flow path which is on the macroscopic scale but in reality as in micro point of view, microscopic point of view if you look into this and this is the path which actually which is subtended by the flowing water. So this is called this microscopic length is called or this path subtended by the actual path subtended by the flowing water is called winding path or the tortious path. This length is called the or this particular ratio of this length to this microscopic length is called the tortiostic factor. So water does not flow from A to B if there is the total head or energy is actually same. If water can flow from A to B if there is a difference of head only that means that if the head at A is higher than head at B then there is a possibility that water flow can take place from A to B. Similarly if the head at B is higher than head at A then water can also flow from B to A. So this path subtended by the actual path subtended by the flow of water, the path subtended by the flowing water in the soil solids is known as the tortious path and the ratio of the tortious length of the tortious path to the length of the macroscopic length or a length of flow path which is called as a tortiostic factor. We have actually studied the Bernoulli's equation and we have defined this Bernoulli's equation as total head is equal to pressure head plus velocity head plus elevation head. So in geotechnical engineering we also use this the total head at a point in water under motion can be given by the sum of pressure, velocity and elevation heads. So PW by gamma w represents the pressure head and which actually has got the units of the length and v square by 2z is the velocity head and v is the velocity with which the water is actually flowing through the soil and z is the elevation head. If you look into this particular application of this Bernoulli's equation in soil mechanics is concerned the velocity head is actually neglected that means that the v square by 2z the kinetic or velocity head also has the units of the length. But since the water flowing velocities typically has very, very small the kinetic head or the velocity head is typically negligible compared to that of the pressure head and velocity heads. For this reason the velocity head is neglected in soil mechanics and jet represents the elevation with respect to the arbitrary datum the datum when we need to if you have a datum and the jet represents the elevation with respect to an arbitrary datum. So the value is the distance of a point at which the head is being measured above the datum. So this can be either positive if the point is above the datum it can be negative if the point is below the datum. So therefore the total head can be pressure head plus elevation head. So water flows in soils only when there is a gradient in head h that means that a difference of head between that is when it is available then there is a possibility of the flow and the lack of a gradient is the head implies that water is not flowing that means that if there is a inadequate head difference h then that means that the water is actually not flowing. Whenever there is a water flow in soils there is energy dissipation occurs. Suppose when there is a water flow in soils the energy dissipation occurs. So in soils water or permeant always flow down the gradient that is water flow from higher energy region to the lower energy regions. So the flow water flow in soils only possible with the possibility of the gradient in head h and energy dissipation and then water flow takes place. So in soils basically the water or permeant which is always flows down the gradient and that is water flow occurs from higher energy regions to lower energy regions. And in this slide example a slide which is actually shown a typical cross section having two soil profiles with a groundwater table which is located at a certain depth below the ground surface and two points which are one and two and one is actually at a z1 meters above the reference datum and is measuring a head of h1 and a pressure head of h2 is measured for a 0.2 which is z2 meters above the reference datum. The thickness of the soil layer which is under discussion is about h units. So the total head here at 0.1 is nothing but the pressure head plus elevation head which is nothing but h1 plus z1 and here total head here is that pressure head plus elevation head that is h2 plus z2. So here as h1 plus z1 is equal to h2 plus z2 is equal to h and the total head difference is actually 0. So though you know there is a difference in pressures here pressure heads which are actually measured here the flow will take place either from 1 to 2 or 2 to 1. In another example which we have discussed in the previous module about the caplarity situation let us assume that we have got the heads in the static water in caplarity tube. We actually have discussed that this particular phenomenon which actually can occur because of the in case of real soils the soil solid water, air solid water interaction because of the air solid water interaction. So here if you have got a groundwater table here at the water level and which is actually if it is considered as a datum and that point is say 2 and hc is the head over which that raise of water has actually taken place because of the caplarity phenomenon. So that point b1. So let us consider here elevation head for the point 1 is hc units above this point 2 and pressure head here as the pressure head is actually negative and which is actually minus c. So the total head is nothing but hc plus minus hc the pressure head is negative here because of the total head is equal to 0. In case of this situation here where you have got elevation head is 0 and pressure head is 0 so total head is 0. So therefore the caplarity flow is not a typical flow situation there is no flow of water in this situation which is actually occurring because of the air soil solid and water interaction. So this is actually shown fluid at rest in soil which is shown in no flow condition and here this is the zone of the capillary raise if the soil is actually completely saturated it actually maintains that negative pressure here but there is a partial saturation there is a possibility that this pressure actually drops down to close to 0 and then the dry soil will actually will be there above this level above the caplarity ground water table level and this is the ground water table level and below this there is a possibility of hydrostatic water pressure that is pwz is equal to gamma wz. So as the depth increases the water pressure increases below the ground water table. At this point at the surface of the ground water table the pressure of the water is equal to 0 because it is open to the atmospheric surface. So in this particular slide what we have seen is that we have got a confined aquifer which is shown with a reference datum and which is actually having a total head of H1 and on the left hand side and total head of H2 on the right hand side. So from the reference datum if you look into this and this particular point if it is named as 1 and if this termed as 2 then this point is nothing but H1 is equal to p1 by gamma w that is nothing but the pressure head plus elevation head at 0.1 for 0.1 that is z1. So p1 by gamma w plus z1 and in this case it is p2 by gamma w plus z2 which is nothing but the total head at H2. So if you observe that and if H1 is greater than H2 that means that water flows down the hydraulic gradient from 1 to 2 that means that the water flows down from here to here. So it is actually with H1 greater than H2 water flows down the gradient. So the difference between these two points difference of the heads between two points here 1 and 2 which is actually shown here which is actually called H1 minus H2 and which is called as the head drop or head loss. So fluid flows down the hydraulic gradient not necessary downhill. So fluid actually flows down depending upon the way the presence of the higher head and lower heads prevalent in the real situations. The in flow of water through soils the main important assumptions which are actually worked out are the soil is actually assumed to be fully saturated and the boundaries between the solids are assumed to be frictionless boundaries and the flow is actually laminar that is in case of soil mechanics particularly flow of water through the porous media or soils. The Reynolds number which is actually defined as rho w into v into d10 by mu w where d10 is nothing but the effective particle size mu w is nothing but the dynamic viscosity of water and v is the discharge velocity the water velocity which is actually through that macroscopic length or macroscopic area and rho w is nothing but the mass density of the flowing fluid. So for a flow to be laminar in case of you know soil mechanics it is actually maintained as r e less than 1 maximum the upper bound value is about 10 or so. So in this figure the cross section a typical cross section is shown here where we have got a datum here and a point a and a point b we are actually interested in flow of water from a to b. So the point a is having a elevation head of z a and a pressure head of p a by gamma w. In this case point b having elevation head of z b and pressure head of p by gamma w equal to h b. So p by gamma w so the total head is equivalent to p b by gamma w plus z b which is actually nothing but small h b which is indicated and here total head is nothing but p a by gamma w plus z a which is indicated here as h suffix a. So the difference in head between point a and point b over a length l which is if you to observe it here h a minus h b which is actually called as a head loss or head drop over a length l over a length l and the soil actually flows the cross sectional area which is actually open to the flow is this one. So this is actually along the length of the along the in the direction of the flow and this is perpendicular to the direction of the flow. So if this is actually having a say for example the entire length for example here if this is actually having a unit cross sectional area a then this length l then that means that the flow is actually occurring over a volume of a into l which is also called as a fluid phase or which the flow is actually occurring. So the head loss between the two points can be given by delta h is equal to h a minus h b which is nothing but p a by gamma w plus h a z a minus p b by gamma w plus z b. So the head loss delta h can be expressed as in a non-dimensional form by using a definition called hydraulic gradient. Hydraulic gradient is equal to delta h by l where i is nothing but hydraulic gradient l is the length of the flow or which the loss of head occurred. So there is you know a relationship between velocity and hydraulic gradient where we have defined in the previous slide as high is nothing but the non-dimensional form of hydraulic gradient over a which is nothing but a hydraulic gradient and which is defined as head loss over a length l. So this particular slide shows the variation of discharge velocity with hydraulic gradient on the y axis what we see is the velocity on the and then on the x axis what we see is the hydraulic gradient i and it can be seen that it is divided into three zones the zone one and zone two and zone three and zone one is actually termed as laminar flow zone it appears that you know this v is actually proportional to i in this region and it exhibits a linear relationship and then beyond zone one once this zone one boundary is cross it is actually termed as zone two which is called as a transient zone and beyond zone two which is actually termed as zone three which is called the turbulent zone. So as the i increases gradually that is hydraulic gradient increases gradually the flow remains in laminar in zone one and zone two mostly and v bears a relation linear relationship with i. So what we have understood is that the flow remains in laminar in zone one and zone two and v bears a relation linear relationship with i at higher hydraulic gradients the flow becomes turbulent that means that at higher hydraulic gradients when the hydraulic gradients are high here the flow is actually becoming turbulent. So in most soils v is proportional to i especially in gravel and very coarse sands the turbulent flow conditions may exist and v is proportional to i is not really valid. So in most soils v is actually proportional to i and the laminar flow conditions are actually established and they are satisfied but in some gravelly soils or the soils which are actually having very large particle sizes or coarse sands turbulent flow conditions may exist. So this particular relationship between the discharge velocity and hydraulic gradient was given by Darcy way back in 1856 and this is a simple equation for which the discharge velocity of the water through saturated soils can be expressed by a relationship known as v is equal to k i which is popularly known as Darcy's law and where k is the coefficient of permeability the units are meter per second which is also called as Darcy's coefficient of permeability this is the one which is actually can be measured in either in the laboratory or in the field and v which is nothing but the discharge velocity or superficial velocity which is the basically the quantity of water flowing in unit time through a unit cross section area of soil at right angles to the direction of flow. So the flow is actually through the pore spaces in soil and not through the cross section area because of that you know the discharge velocity is basically is called as a superficial velocity and this is formulated based on the observations of flow of water through clean sands. This particular Darcy's law which is actually has been in a formulated based on the observation through flow of water through clean sands and further the Darcy's equation is combined with a continuity equation which is nothing but q is equal to v into a and a is the cross sectional area over which the flow is occurring and which into multiplied by v we get discharge that is q is equal to v into a is equal to k i a where k is the coefficient of permeability and delta H by L is nothing but the hydraulic gradient that is head loss over a length L and a is the area over which the flow is actually happening. So q is equal to total rate of flow through the cross section area a and k is the Darcy's coefficient of permeability. This Darcy's coefficient of permeability is defined as a property of the soil and which is actually defined as ease with which water can flow through the soil that means that some soils allow the water flow very easily and some soils very difficult to water for the water to flow through the some soils that means that you know this particular property is used in many ways in constructing many geotechnical structures. So here in this particular slide a new term called seepage velocity is introduced the seepage velocity is actually velocity which is actually happening through the voids that means that if you consider you know a cluster of soil particles and water which is actually filled in the voids and the water and the water flow is occurring from say in this case from top to bottom and what actually happen if you take a magnified view of the distance between soil particles. So if this is actually happen to be a and this is actually nothing but the that is area of the voids. So here before entering the velocity is v and here the velocity which is actually within the voids is called as the seepage velocity again once it crosses the void is actually discharge velocity. So here because of the interconnected to the voids through the soil only the seepage velocity the flow of water through the soil is you know which is actually indicated by a seepage velocity with v suffix s. Suppose if this is actually indicated as a two phase diagram as is actually shown here then we have got water and soil solids and the total area of this is that area of water area of the water that is nothing but the AV and the area of the solids and which is nothing but the A is equal to AV plus AS and from the for the unit width of the sample if you take and void ratio which is nothing but the volume of voids to volume of the solids with that we can write as AV by AS and using the principle of continuity Q is equal to V into A and for A this continuity equation now we can write it as V into A that means that here V and then area which is actually available here is A and when it comes to VS here the width got reduced now AV which is nothing but the VS into AV. So by using this VS is equal to we can write it by A by AV into V and this for a unit width of the sample we can write it as V by volume of voids which is nothing but you know from if you take the definition of the porosity we can write it as VS is equal to V by N. So the seepage velocity is actually nothing but 1 by N times the discharge velocity. So as the porosity cannot be more than 100% VS always greater than V, VS is always greater than the seepage velocity always greater than discharge velocity. So as the as a particle of water proceed from A to B it exerts a frictional drag on the soil particles. That means that when the particle when the water is actually flowing from A to B because of the some head difference which is actually happening between A to B the water actually flows from A to B and in the this is in that this is called the direction of the seepage. So when the water flow is occurring from A to B as a particle of water proceeding from A to B it exerts a frictional drag on a soil particles. So in turn it produces this so called frictional drag which produces a seepage pressure in the soil structure. Sometimes if the seepage pressure is excessive then it can lead to a failure in a geotechnical structure. So seepage pressure is due to flow of water through the voids. When the water is actually flowing through the voids because of the frictional drag exerted by the flowing water on to the soil solids the seepage pressure is actually generated. So because of the frictional drag the hydraulic head decreases steadily on every flow line. That means that as it actually drops down from A to B because the energy is actually transferred in the form of exerting a frictional drag on the soil particles the head actually decreases steadily from say from A to B in this example. And we actually have discussed that the total stress which is nothing but the effective stress plus pore water pressure. And we also have discussed this for a case where no flow condition that means that there is no difference in head between two points in the consideration. In that case you know we say that hydrostatic pressures provide. So changes in geostatic stresses, geostatic stresses are nothing but when the earth surface is horizontal and having a unit weight which is actually more or less constant along the in all directions. And the change in geostatic stress with the flow of water the soils if you look into this. When water flows through the soil it exerts a drag force called seepage force that we have actually discussed. And the presence of seepage force which causes the changes in the direction of flow which causes changes in the pore water pressure and the effective stresses in the soils because of that the changes actually happen. So in this particular slide three cases which are possible one is that when no flow takes place through the soil a hydrostatic condition. So here we have got a soil which is saturated and having a thickness H and H1 is the head of water here which is maintained constant. And this limb which is almost at the same surface then what we can actually see is that the total stress is nothing but gamma WH1 plus gamma WH1 plus gamma sat H. And the pore water pressure is nothing but as the water is up to the surface and soil is completely saturated. So it is nothing but the gamma WH1 plus gamma WH1 into H1 plus H. So here the effective stress is nothing but total stress minus pore water pressure with this what will actually happen is that here we have got gamma dash H that is here this particular condition is a no flow condition. And it also tells that the head loss is actually delta H is equal to 0 because both at this point and this point that the total head is actually is 0 or the same. So because of that what will happen is that the head loss is actually 0. And another thing is that if H1 is say for example even if it is 1 meter, 2 meters, 3 meters or 5 kilometers for example the changes in the effective stress is not there. That means that it is independent of the thickness of the height of water above the ground surface. If the ground water surface for example say within this H then it can actually increase or decrease depending upon the changes which actually can happen. For example if there is a depletion of water table the effective stress increases. If there is an increase in water height there is a decrease in the effective stress. So here if there is a excessive increase in the effective stress and that also can lead to the crushing of soil particles and then the collapse of the soil. And if there is a excessive decrease in the effective stress that also can lead to the problems which we are going to discuss. In case 2 here what we did is that when the flow takes place to the soil and we are actually referring here what we did is that we took this limb down and that is below the surface. And where we say that when this situation is there water flows from through the soil from upward in the downward direction here. So this is the flow of water which is actually happening to the soil. And this H which is either difference between water levels between this surface and this surface and which is actually below this surface. And for this case if you look into this the total stress remains same that is nothing but gamma w H1 and the soil is actually saturated here again gamma w H1 plus gamma sat H. But if in case in the previous case 1 when there is no flow of water then we actually have said that you know the pore water pressure is like this that is gamma w H1 plus H. But now here what is actually happening is that the water which is actually flowing from top to bottom here that means that here the there is a head which is actually available at H units is available. And by the time actually it comes out of this the available head is actually 0 for the water to flow through the flow. That means that over a length here so here the hydraulic gradient is nothing but I is nothing but small h by capital H. And at the beginning of this point the point where the soil surface is actually commences where the head available is H and at this point the head available is 0. At this midpoint let us say h by 2 units from here it is 0.5 times the small h is actually available here the 50 percent of the head available that means that 50 percent already you know got dissipated or spent in making the water flow from top surface to the middle surface here. So because of that reason there is a drop in the or decrease in the pore water pressure because of the loss of this head. So the pore water pressure at point at this point is nothing but gamma w plus H1 plus H minus H. So that makes actually the pore water pressure because when the water flow takes place from top to bottom at this point it is gamma w into H1 plus H minus small h. So with that now again if you take the effective stress which is nothing but the total stress minus pore water pressure at this point there is no change because the gamma w H1 but here what actually happens is that the pore water pressure when there is no water flow we have got effective stress as gamma 1 H but now because you know these two terms will get cancelled now what we will have is that gamma 1 gamma dash H plus H gamma w that means that here when water flows from point top to bottom that is in downward flow there is an increase in the effective stress. And if we replace that H as I times H we can actually say that gamma dash H plus I times capital H gamma w where H is the thickness of the soil layer over which the flow is actually happening and gamma w is the unit weight of water. Now let us consider a case where case 3 when the flow takes place to the soil upward flow that means that what we have done is that this limb actually has been taken above this water surface and this water surface is actually maintained and this level or head is actually maintained here. So when these are connected here and what actually flows from this point this end to this end. So this upward flow if you look into this decreases the effective stresses in the soils the total stress now if you look into this again it remains same gamma w H1 and gamma w H1 plus gamma sat H but here if you look into this the pore water pressure without any flow without any flow is nothing but gamma w H1 plus H but now what is actually happening is that the head available here is small H the head available here is 0 that means that the hydraulic gradient over this length is that nothing but I is equal to small h by capital H and because of this what is actually happening is that the pore water the head available at this point is nothing but gamma so the pore water pressure ordinate here is gamma w into H1 plus H plus small h and here it is gamma w into H1. So because of this what will happen is that if you take the difference of this and this because of the this particular condition what is actually explained here with that the effective stress is nothing but gamma dash H minus H gamma w when I when we replace that H as I times capital H capital H then we can write this effective stress at this particular point as gamma dash H minus H gamma w that means that when the water flow takes place from in the downward direction we have seen that effective stress increase when the water flow is taking place in upward direction this particular situation is actually possible in the real conditions like in attrition conditions where you have got a more head of water than the you know hydrostatic head which is actually existing there or it actually has got a source of water and that particular previous layer which is actually beneath a impervious layer in that situation this type of upward flow conditions are more common. So this gamma dash H minus H gamma w where in upward flow causes the decrease in the effective stress. So what we have understood is that seepage is the flow of water through the soil and it is a basically it is a phenomenon which actually happen in soil and basically it exerts a frictional drag on soil particles and which is actually called as seepage force which results in a head loss and the seepage force play a very important role in destabilizing geotechnical structures and the seepage force per unit volume is actually called as seepage pressure say for example here the downward seepage increases the effective stress that is what we have actually said sigma dash is equal to gamma dash H plus i gamma w into capital H. So i gamma w when I replace it with PS then what we can say that downward seepage increases the effective stress and upward seepage actually decreases the effective stress that is nothing but sigma dash is equal to gamma dash H minus PS H where PS is equal to i gamma w where seepage pressure which is actually having a units of kilo Newton per meter cube and PS is indicated by i gamma w. So if you take a hydraulic structure the water is actually retained at the upstream and downstream so here when the water actually flows from A to B along this hydraulic structure at this here in this portion there is an increase in the effective stress and in this portion particularly there is a decrease in effective stress. So the decrease in the effective stress particular phenomenon causes you know situation where there is a possibility of you know endangering the stability of a particular hydraulic structure. So on the right hand side what we see is that seepage forces decrease the effective stresses reduce and the resistance to the embedment also decreases. So the stability of a structure is actually highly depend upon this particular portion where there is a you know upward flow occurs. So seepage pressure seepage stresses play a key role in reducing the stability of a geotechnical structure retaining water or any water body. The hydraulic gradient at which the effective stress becomes 0 is called as the critical hydraulic gradient. So here we are introducing a term called critical hydraulic gradient previously we said that hydraulic gradient is nothing but i is equal to h by L and h is nothing but a head loss or a head drop over a length L but if the hydraulic gradient at which the effective stress is becoming 0 is known as the critical hydraulic gradient i suffix c. So for example here we have discussed in the previous slides that sigma dash effective stress is equal to gamma dash h minus i gamma w that is nothing but with gamma w h is equal to 0 with that under these circumstances when i tends to ic the effective stress actually becomes 0 that is gamma dash h minus gamma w h is equal to 0. Now these are circumstances the cohesionless soils particularly the soils without any binding capacity cannot support any weight. So when this situation of critical hydraulic gradient conditions exist the soil loses its supporting capacity. Moreover as i tends to ic the soil becomes much looser and permeability of the soil increases drastically that means that when i tends to ic soil becomes much looser and the permeability of the soil increases drastically. So here we are actually defining in this particular side a condition called quick condition or boiling condition in cohesionless soils. The particular situation which is actually shown here is that there is a point B and then there is a point A and point A is at the elevation that is the datum and total head at A is equal to h and total head at B is equal to small b and here what we can see is that we have got a head of water and which is nothing but here that is h minus b and total head if you look into this pressure the head is nothing but the h minus b plus b that is h units. So at point A the pore water pressure at point A is gamma w h and at point B that is this is at this point here. So at A that is sigma v dash that is nothing but gamma sat B that is the thickness of the soil layer into the saturated weight of the soil into b minus gamma w h. So when i is nothing but h minus b is nothing but the head loss over a length of the soil that is b. So we can simplify that h is equal to b into i plus 1 by writing i is equal to h minus b by b and by simplifying we can write h is equal to b into i plus 1 and by substituting in sigma b dash is equal to gamma sat into b minus gamma w h we can by substituting we can write gamma sat b into minus gamma w b into i plus 1. So we have written h in terms of b and i. So for quick condition to takes place sigma v dash has to be 0 that means that i has tends to become ic. So with that what will happen is that ic is equal to gamma dash by gamma w which is nothing but gs minus 1 by 1 plus e or gs minus 1 by specific volume that is 1 plus e is indicated as specific volume. So ic is equal to gamma dash by gamma w which is nothing but gs minus 1 by 1 plus e. So quick send there is a terminology which is actually comes when there is a the loss of the effective stress actually happens drastically because of the phenomenon what we discussed is that when the water flow happens the soil in the upward direction the soil loses all its strength and then it loses the supporting capacity and we actually have got a phenomenon called quick send phenomenon. So quick send phenomenon is basically a condition where water flow occurs in the upward direction and then there is a possibility of the you know the loss of the effective stress. So one should note that quick send is a is not a type of sand but a flow condition occurring within cohesion soil or a soil which are actually not having any adequate binding capacity when it is actually effective stress is reduced to 0 due to upward flow of water. So quick send occurs in nature when water is being forced upward under pressurized conditions. So in this case the pressure of the escaping water exceeds the weight of the soil and the sand grains are forced apart and the result is that the soil has no capability to support a load. So why does actually quick send condition or boiling occurs mostly in fine sands or sills basically is because of the you know the binding capacity which is actually there. So some practical examples of the quick conditions are that excavations in granular materials behind the coffer dams along side rivers and any place where the artesian pressures exist that is where the head of the water is greater than the usual static water pressure. So this phenomenon what we said we have actually also said that when there is you know higher head of water than the you know in the soil beneath because of the either because of the water source or because of the some previous layer beneath the impervious layer. So in that situation the artesian conditions may exist and when a suppose if the excavation is actually made in such type of soils then they can actually lose to the subjected to a phenomenon called boiling condition and particularly it can lead to a failure. When a previous underground structure is continues and connected to a place where the head is higher. So and sometimes when behind the river embankments when we are constructed or levees which are constructed to protect the floods. So the quick conditions are actually popular particularly in excavations in granular materials behind the in the river bed particularly with sandy soils and any place where artesian pressures exist and if the soil is actually you know is displaced because of the excavation because for installation of the utilities are some other reasons. So in this particular lecture what we actually have introduced ourselves is that this permeability and seepage one wherein we actually have discussed that you know how and how important is water flow through the soil particularly in the application and design of the geotechnical structures and we also have defined a term called hydraulic gradient and also a term called velocity and seepage velocity and seepage velocity what we said is that is with a relationship vs is equal to v by n and seepage velocity is always more than v because seepage velocity is the velocity which is actually happening through the voids of the soil and then we introduced a term called hydraulic gradient which is nothing but i is equal to h head loss over a length l then we also have introduced a term called critical hydraulic gradient particularly i is equal to defined as hc by l and where hc is nothing but head loss at a particular place where the effective stress is tend to become 0 and we also have discussed in the three cases one a no flow condition where total head is actually difference is 0 and one condition is that downward flow in that case what we have discussed is that when the water flows upward downward there is an increase in effective stress when water flows upward because of the certain prevalent conditions and it can actually lead to the decrease in effective stress. One practical example what we discussed is actually a quick sand condition and we said that quick sand is actually not a type of sand it is a condition of flow condition where water flows which is actually arising because of the water flow in the upward direction.