 Welcome to lecture number 10 module 1. In this lecture we are going to introduce ourselves to concept of effective stress and capillarity. In the previous lecture we tried to understand about compaction of soils and factors affecting compaction and methods for the deep compaction. This particular concept of effective stress has lot of prominence in geotechnical engineering. So the title of the lecture module, this lecture is Effective Stress and Capillarity. So as you can see in this slide a cross section of soil profile is shown with ground water table. So the zone below the ground water table that is the soil below the ground water table is completely saturated with water filling all the voids within the soil. So here this particular zone here it is shown that a partially saturated nature where you have got air and also a part of water. So mostly soils are actually completely saturated below water table and above water table the soils are partially saturated in nature. So we have some phenomenon depending upon the type of soil a capillarity phenomenon which occurs above the ground water table. So here the water surrounding particles at points of contact between particles and filling the small void spaces are shown here. So this particular portion which is within the soil solid space is called pore space or this is indicated as a small d. It is pore diameter or it can also be called as pore radius. As we can see from this slide soil can be visualized as a skeleton of solid particles enclosing continuous voids which contain water and air or air. So soil skeleton which individual soil grains and the arrangement which is actually shown here and the volume of the soil skeleton has a whole can change due to rearrangement of soil particles into new positions. The new positions they acquire mainly by rolling and sliding with a corresponding change in forces acting between the particles. So the soil can be visualized as a skeleton of solid particles enclosing continuous voids which contain water and air. If you look into in a fully saturated soil since water is considered to be incompressible a reduction in volume is only possible if the water can be escaped from the voids. If the water can escape from the voids. In case of a dry or partially saturated soil reduction in volume is always possible due to the compression of the air in the voids provided there is a scope for particle rearrangement. So in dry or partially saturated soil a reduction in volume is always possible due to compression of air in the voids provided there is a scope for particle arrangement. Before introducing the concept of effective stress we have to introduce ourselves to what is total stress. This vertical subsurface stress resulting from the soil mass because of the self weight alone is actually shown here. Suppose if this is the portion of the ground surface and if you consider the unit area and extending up to depth z the sigma v is nothing but weight of the soil in this zone divided by the unit area. So if gamma t is the average unit weight of the soil in this zone gamma t into the volume of this particular element divided by the cross section area which is unit area so with that sigma v is equal to gamma t z. So gamma t is nothing but the unit weight of the soil homogenous from the ground surface to the depth z. So sigma v which is also if the ground surface is horizontal which is actually called as geostatic vertical stress. So sigma v is equal to gamma t z and is also referred here as total stress which is resulting at a given point because of the self weight of the soil. When pore water pressure or pore pressure is the pressure in the water in the void spaces or pores which exist between and around mineral grains. So here schematically arrangement of the grains are shown here and this is the pore water pressure exerted in all directions along the inner surfaces of the soil solids can be seen from this slide. So this is the u pore water pressure which has multidirectional orientation. So pore water pressure is the pressure in the water in the void spaces or pores which exist between and around the mineral grains. So pore water pressure under no flow conditions is also given by the hydrostatic pressure. If you assume that there is a ground water table and if this is the point where the pore water pressure is say tapped through a stand pipe then h is the height of water in the stand pipe. The pressure is nothing but u is equal to gamma w into h, gamma w is nothing but the unit weight of water. As the name implies this is the pressure which exists in the water which is present in the pores of the soil that is the pore water pressure as the name implies is the pressure which exists in the water which is present in the pores of the soil. So the soil pores are normally interconnected and they may be visualized as being a highly intricate and complex correction of the irregular tubes. So if I have a straight tube even then that the pressure is same and if you have got an interconnected void which is of this shape or this shape or this shape the pressure is actually is nothing but the water pressure which is actually connected through along the voids. So the soil having interconnected voids which are similar to this is analogous to irregular tubes connecting the or passing through the voids. The size of the tubes depends upon the size of the pores within the soil. So the effective stress principle the way back in which is given by Carl Terzaghi in 1936 is valid for only saturated soils this effective stress sigma dash so effective stress is indicated by sigma dash and which is at a point in a soil mass is equal to total stress sigma at that point minus the pore water pressure u at that location. So sigma total stress is apportioned by sigma dash effective stress and pore water pressure or effective stress is nothing but sigma minus u w both total stress sigma and pore water pressure u are physically meaningful parameters and stresses that can be actually measured in the field. So effective stress sigma dash can only be computed and from the measurements made from the total stress and pore water pressure. So the effective stress sigma dash at a point in a soil mass is equal to total stress sigma at that point minus the pore water pressure u at that location. Terzaghi in 1936 proposed the relationship for effective stress and what he quoted is mentioned here all measurable effects of a change of stress such as compression distortion and a change of shearing resistance or due to changes in effective stress. All measurable effects like change of the stress such as compression in the soil or distortion or change of shearing resistance or due to the changes in the effective stress. So certain aspect of the engineering behavior of the soil especially compression and shear strength or functions of effective stress which we will be discussing these aspects in forthcoming modules. So first this is Terzaghi's concept of effective stress is the first important equation in geotechnical engineering. So effective stress by definition now what we discussed is that can be determined only by arithmetic manipulation that is sigma minus u w. Like sigma and u w sigma dash is thus not a physical parameter it is only a mathematical concept but obviously it is a useful parameter since it has empirically been observed to be determinant of the engineering behavior of the soil. So though it is cannot be measured it actually has got lot of physical significance and has empirically been observed to be determinant of the engineering behavior of the soil. So effective stress concept through an idealized and saturated element under stress can be considered like this. So here in this particular slide a cluster of soil grains are shown and A dash and A dash is the line passing through the voids. So A dash and A dash is the line passing through the voids and this is something like a wavy shape it can be because the line pores or the line cannot pass through like a straight line. So this is the pore water and sigma is nothing but the total stress and now here we knew that sigma total stress is equal to pore water pressure and effective stress. So this is an idealized saturated soil element in the equilibrium and wavy plane A dash and A dash passes through the particle to particle points. So wavy plane A dash and A dash passes through particle to particle contact points almost entire plane passes through the pore water here you can see here this one. So here you consider you have got a imaginary line A dash and A dash is a stretched view of plane A dash and A dash. So A double dash and A double dash is the stretched view. So here you can see AC is the area of the contact between the soil grains and AW is the area of the water. So total area is equal to AC plus AW and AC on the this is the area of the contact where in the force F dash is actually acting on the granular matrix and so UW is the water pressure acting in the area AW. So further once you extend this one, so here sigma into A is the total stress and account of overburden. So sigma multiplied by the area of the plane A. This is nothing but the force which is acting because of the sulphate of the soil above that plane A dash and A dash and A dash is the imaginary line which is the representation of the wavy line connecting the particle to particle contact points. Since the soil element is in equilibrium the algebraic sum of the forces must be equal to zero and UW AW is the pore water pressure multiplied by the area of the plane which passes through the pore water. So AW is nothing but the area which is passes through pore water and F dash is nothing but the summation of the forces which act at particles to particle contact through which the plane passes. Let us say that you have got say you know 100 number of particles then F1 plus F2 plus F3 plus F100 is equal to F dash. So applying the laws of statics soil element in equilibrium we can write that AC plus AW is equal to A. So with this we can write sigma into A is equal to F dash plus UW into AW. So sigma is equal to by taking A this side we can write F dash A plus UW into AW we can write it as A minus AC is nothing but area of the water that is area occupied by the pore water is nothing but the A minus AC. So I can write UW into A minus AC by A so sigma is equal to sigma dash plus 1 minus AC by A into UW. So I can write here sigma dash that is sigma dash plus 1 minus A into UW is equal to sigma. So A is nothing but contact area between the particles per unit gross area of the soil that is nothing but A is nothing but the ratio of contact area between the grains to the total area which is under consideration. In granular materials because the contacts will be like minute contact points like point point contacts so A tends to be 0. In that case sigma is equal to sigma dash plus UW. So effective stress is not the stress at particle to particle contact and the stress at particle contact is physically the stress equal to F dash by AC, AC is nothing but the area of the contact between the soil solids. So here with this what we have deduced is that we have deduced the equation which is proposed by Terzaghi that is sigma is equal to sigma dash plus UW for granular materials with where A tends to equal to 0. For when with the area of the contact into consideration we can write sigma is equal to sigma dash into 1 minus A into UW. Now in this slide further extending to the effective stress concept in here consider you have got this wavy line passing through this point and this is the line which is actually imaginary line which is extended and so at point O that is at this point O the water table is here. So the total stress at this point is nothing but gamma H1 that is dry soil into this saturated soil because the unit weight will be saturated being gamma sat into H2. So the total stress is gamma H1 plus gamma sat H2. Water pressure is nothing but UW is nothing but gamma W H2. So effective stress at point O is nothing but gamma sat H2 plus gamma H1 minus gamma W H2 with that I can write sigma dash is equal to gamma H1 plus gamma sub or gamma dash H2. So this is you know how we can actually determine vertical stresses, effective stresses by knowing total stresses and pore water pressures. So effective stress is sometimes interchangeably used with intergranular stress although the terms are approximately same there is some minor difference. Total vertical stress F at the level of O that is the point O what we have seen in the previous slide is the sum of the following forces. One is that forces carried by the soil solids at their point of contact that is Fs. So Fs is equal to F1 vertical that is nothing but the vertical component of each contact force we are taking F1v plus F2v so on to F3v so on to Fnv and forces carried by water which is nothing but Fw is equal to UW into A minus Ac. Electrical attractive forces between solid particles at the level O that means that if you have got say some soil solids which actually have got mineral interaction then electrical attractive forces will be there between the solid particles at the level of O that is Fa. Electrical repulsive forces between solid particles suppose if they are dispersive in nature then there can be a dispersion that is Fr. So we have net to forces acting on the grains is that one is due to the contact other one is forces carried by the water and other one is forces due to attractive forces and another is that repulsive forces. So the total vertical force we can write it as Fs is equal to Fs that is due to contacts plus Fw carried by the water minus Fa plus Fr. So here sigma we can write as by dividing by A sigma Ig which is intergranular that is Fs by A plus UW into 1 minus Ac by A minus A dash plus R dash. So sigma is equal to sigma Ig plus UW into 1 minus A minus A dash and R dash where sigma Ig here is nothing but intergranular stress and Ac is equal to A is equal to Ac by A that is the contact ratio and A dash is nothing but the electrical attractive force per unit area of the cross section R dash is electrical repulsive force per unit area of cross section of the soil. So A dash and R dash are the electrical attractive forces or repulsive forces per unit area of cross section. So we can write here sigma Ig is equal to sigma minus UW into 1 minus A plus A dash minus R dash. So the value of A is very small in the working stress range because of the minute contact points so A tends to be equal to 0. So with that we can write sigma Ig is equal to sigma minus UW plus A dash minus R dash. So for granular soils and sills and clays of low plasticity for granular soils and silty soils or clays having low plasticity the magnitudes of A dash and R dash are very small. So for all practical purposes the intergranular stress becomes effective stress which is nothing but sigma dash is equal to sigma minus UW. If you have highly plastic and dispersive clays then A dash minus R dash is large. Such situations we cannot say that sigma Ig that is intergranular sigma Ig which is nothing but the effective intergranular stress is not equal to sigma minus UW because here in clay soils mineral crystals are not in direct contact since they are surrounded by the adsorbed water layers. So here the platelet particles are shown with the adsorbed water layers surrounding adsorbed water layers are shown here and here this is the phase to edge attraction. So phase to edge attraction you can see here edge to the phase attraction of the clay platelets is shown here. So this type of configuration here in clay soils mineral crystals are not in direct contact and since they are actually surrounded by the adsorbed layers. So it is assumed that the intergranular forces can be transmitted through the adsorbed water layers. So intergranular soils are assumed to be transferred through the adsorbed water surrounding the platelet particles. So we have seen that the concept of effective stress is valid only for saturated soils. Nowadays lot of work is actually happening on unsaturated soil mechanics. So the effective stress in a partially saturated soil can be defined like this. The partially saturated soils they are basically three phase state the water in the voids is not continuous and pore air occupies the considerable volume in the system. So here you have got water and pore air within the, so in the pore space you have got the pore air as well as pore water. So you have got two components one is air another one is the water within the voids. So these are the solid particles and this is the pore water which is actually shown here. So total stress at any point is nothing but effective stress plus pore air plus pore water pressure. So based on that discussion we can actually say according to Bishop 1960 sigma that is total stress is equal to sigma dash plus UA minus psi into UA minus UW. So psi is the fraction of the unit cross sectional area of the soil occupied by water. For dry soil degree of saturation is zero the psi is equal to zero. For saturated soil psi is equal to one that is for degree of saturation is equal to 100%. So for intermediate values the of SR psi is actually read from the chart which is actually given by Bishop from the triaxial test by conducting triaxial test the values of psi that is the fraction of the cross sectional area of the soil occupied by the water. So Bishop 1960 determined the nature of the variation of psi with several degree of saturation of several soils based on the triaxial test for the unsaturated soil specimens. So in this slide this particular graph which is actually gives variation of degree of saturation with this psi so as can be seen is that with increase in degree of saturation the value of the psi tends to become one the effect of fluctuations of water table on the effective stress. So here in this particular slide a soil strata which is actually shown here a saturated soil strata and the vertical stress at this point sigma v and at this line that is at this point where z is equal to h1 plus h that is at this point sigma v and sigma v dash. So we need to determine sigma v and sigma v dash along this particular plane so 1 1 is the initial water location at ground surface and 2 2 is the water level during say rain so water level there will be a raise in water table from 1 1 to 2 2. So if you consider a situation at 1 1 initial water level location before rain so sigma v we can give as gamma sat into h u w is nothing but gamma w h so effective stress is nothing but gamma dash h 2 2 water level location during rain sigma v is equal to gamma sat h plus gamma w h1 with that u w is equal to gamma w into h1 plus h so we can say that sigma v dash is again equal to gamma dash h. So the raise of water level above ground surface increased both u w and sigma by the same amount what you can see is that the raise of water level above ground surface increased the u w that is water pressure in the soil and sigma by the same amount and consequently the effective stress is remaining unchanged so hence if the water table is say 2 meter above ground surface or 100 meter above or 5 kilometer above the change in the effective stress and in a saturated soil mass will not be there. Now here the ground surface which is actually shown here and water table is at the ground surface and 2 2 is the water level after rain so there is a depletion of water table. So we need to determine what are the vertical stresses total stresses and effective stresses at this point. So case one again initial water level before rain so as we have done we have got sigma v dash is equal to gamma dash h or gamma sub h. In 2 2 water level after location sigma v is equal to gamma d into h1 plus gamma sat into h minus h1 so with that u w is equal to gamma w into h minus h1 where we have got here sigma v dash is now more than gamma dash h. So depletion of water table causes the increase in the effective stress. So any change in the water level within the soil surface causes the increase in the effective stress. So could lead to sudden depletion of water table let us say that if there is an excavation or if there is a sudden depletion of water table it could lead to the increase in the effective stress. In the increase in the effective stress is nothing but the increase in the integral large stresses and could lead to the high increase in the contact stresses and this lead to the crushing of the grains and consequently this results in the settlement of the adjoining structures if it is a course of excavation. So this is the summary what we have actually discussed with a shift in the water table there is a change in the distribution of pore water pressure with the dip and the time interval is long in soils like clays in which water flows slowly and almost instantaneous in soils like sand in which water level very fast. So when pore water pressure are adjusting to the new location of groundwater table the condition of water can be described as the transient hydrodynamic phenomenon and after achieving equilibrium condition it changes to the hydrostatic conditions. The effect of the fluctuation of water table on the distribution of f2 stress with the depth can be summarized further as follows. For water table below the ground surface a raise of water table causes a reduction in the effective stress and a fall in the water table produces an increase in the effective stress. So if there is a raise in water table it causes a reduction in the effective stress and a fall in the water table produces an increase in the effective stress. So water table above the ground surface fluctuation in the exposure water level does not alter the effective stress in the soil that is what we have actually discussed in previous slides. So during monsoon the ground water table is known to rise and hence effective stresses reduces. So does shear strength so when the effective stress reduces when there is a volume changes when take place in the soil sample so thus the shear strength also reduces. When shear strength reduces below the magnitude of the shear stresses in soil then sliding and collapse take occurs. So increase in sigma occurs instantaneously whereas an increase in effective stress is not instantaneous since particle adjustment and readjustment is not instantaneous. So let us have to based on the discussion whatever we had let us look into this particular example one. So here a soil strata which is actually shown in this slide where you have got a gravelly sand gamma sat is 18.5 kilo Newton per meter cube gamma m is 17 point that is gamma m is nothing but the gamma bulk is 17.8 kilo Newton per meter cube and this depth is 4 meters and this depth below the water table is 2 meters up to the top surface of the sand layer. Sand layer gamma sat is 19.5 kilo Newton per meter cube sandy gravel gamma sat is equal to 19 kilo Newton per meter cube and the depth is 5 meters. So we have a strata of 5 plus 4 9 9 plus 6 15 meters. So we need to plot the variation of trot stress effective and effective stresses and pore water pressure with the depth for the soil profile shown below. So here what we have is that if you take the gamma sat here then what we can actually plot this the plot which is actually shown here. Let us see here this particular variation shows the total stress here. So you can see that this is a point let us say at point 10 meters that is here we need to find out. So here what we use is that sigma v is nothing but the 4 into 17.8, 17.8 is nothing but the bulk unit weight of the soil because here it is partially saturated below the water table same layer it is saturated so we need to take this one. So here what we do is that 4 into 17.8 plus 2 into 18.5 that is the saturated portion. So we are now come to the water table. So water table is at this particular location so 4 into 19.5 that is the another layer. So with that we can actually find out the stress here total stress as 186.2 kilo Pascal's. U that is pore water pressure here it is 0 and then water table starts from 4 meters so it is 0 here and at this point at 10 meters because of the 6 meters that is 10 minus 4 6 into 9.81 which is nothing but the 58.9 kilo Pascal's. So v dash is nothing but this minus this you are actually having this is the profile for the this is how the variation of the effective stress with the depth for the given problem is shown here and this is the variation of the total stress with the depth and this is the variation of the water pressure with the depth 0 here and then it actually starts like this. So then one more interesting issue we should understand is that the pore water pressure is also called as neutral stress. Why and you know this can be demonstrated through a simple illustration here. Let us assume that you have got a soil having certain thickness and it is loaded with lead balls equivalent to the intensity of sigma and if that actually induces a change in void ratio from E0 to E1 the void ratio changes or reduces from E0 to E1. E0 is the initial void ratio or even is that the new void ratio which is because of the loading. So E0 to E1 produces a change in other mechanical properties of the soil so for this reason it is actually called as effective stress. So here only effective stress can induce changes in the volume of the soil and can produce frictional resistance. So here sigma which is surcharge before this is the location this is the level before placing the lead balls and this is the surface after placing the lead balls that means delta E is nothing but the delta H that is the change in thickness over certain age. So delta E which is nothing but the change in void ratio and there is a reduction in the volume also. So the increase in pore pressure due to the weight of the water does not have for let us assume that now the same sigma we have replaced with having sigma by gamma w. So we have maintained hw water the pressure is equivalent to that of the sigma which we have induced in the previous slide. So the portion of the water equivalent to sigma which is nothing but gamma w hw so the increase in pressure due to the weight of water does not have a measurable influence on the void ratio or any other mechanical property. So here this provision of this will not change any effective stress therefore the pressure produced with water is called as neutral pressure. So neutral stresses cannot by themselves cause volume change or produce any frictional resistance that is the reason why because on its own it cannot actually produce any changes and the neutral stresses cannot themselves cause any volume change when volume change is not there that means that there is no change in the effective stress and they cannot produce any frictional resistance. So the increase in the pressure due to the weight of the water does not have a measurable influence on the void ratio or any other mechanical property. Now this capillarity which is a extension to the whatever we have discussed and the reason for understanding capillarity is gaining interest in many environmental geomechanics problems particularly in the issues of contaminant migration and other issues in the ground water particularly zone above the ground water table. So the ground water table or a periodic surface is the level at which the underground water will raise in an observation well pit or other open excavation into the earth. In addition every soil in the field is completely saturated up to some height above the water table and particularly up to some more height and this is attributed to the phenomenon called capillarity in soils. So capillarity is a phenomenon which is actually caused by interaction between soil solids air and water interaction. So capillarity arises from a fluid property known as surface tension which is a phenomenon that occurs at the interface between different materials. So capillarity arises from a fluid property known as surface tension and which is a phenomenon that occurs at the interface between different materials. So for soils the surface of water mineral grains and air. So here the capillarity that surface tension phenomenon basically takes place because of the surface of water mineral grains and air that is the capillarity is resulting because of interaction between surface of water mineral grains and air. So here the definition of surface tension is given for the interest of the students caused by each portion of the liquid surface exerting tension due to molecular attraction on adjacent portions of the surface or on objects that are in contact with the liquid surface. So here we have got you know this capillarity phenomenon causes few issues. One is that capillarity rise, capillary rise which is called there is a rise of water above the ground water table then the rate of rise of water that is the capillarity velocity time and the velocity with which it rises that is called capillary velocity. The phenomenon in which water rises above the ground water table against the pull of gravity but it is in contact with the water table as its source is referred to as capillary rise. So the phenomenon in which water rises above the ground water table against the pull of gravity but is in contact with the water table as its source is referred to as capillary rise. So water is sucked up into the pores of the soil in this zone on account of the surface tension of water. So a manifestation is referred to the capillarity phenomenon. So we have in the capillarity water system the zone of capillary saturation close to the ground water table we can actually and spit 100% saturation and the zone also depends upon the type of soil and zone of partial saturation which where from the capillary saturation to it will be just above the zone of completely capillary saturation above the zone of saturation is a zone of capillary saturation and the above is the zone of partially capillary saturation in this zone the water is connected through the smaller pores but more of the larger pores are filled with air. So here there is a possibility that at this interface the air entry takes place and zone of contact water this is the water in the zone surrounds the points of contact between the soil particles and also surround the soil particles but is disconnected through the pores. So we have three zones one is zone of capillary saturation which is completely saturated and zone of partial saturation that is above capillary saturation and zone of contact water where it is also called as contact moisture between the soil grains. So here in this particular slide for an excavation side suppose if this is the ground water table so this is the saturated with the periodic water and this is saturated with the capillary water and this is the partially saturated zone with capillary water and partially saturated with percolating water only. So here there is a possibility that ingress of rain water can come because of the precipitation of the rain water so this is partially saturated because of the percolating water and this is due to the capillarity phenomenon but it is partially saturated and this is so here the degree of saturation is 100% here then the degree of saturation tends to reduce from 100 to 20% or so because of the presence of the partially saturation that is because of the presence of the air. So capillary rays which is definition a rays in the liquid above the level of zero pressure due to a net upward force produced by the attraction of water molecules to the solid surface. Say for example glass which is if you visualize all the interconnected tubes as interconnected length as glass tube or soil for those cases where the adhesion of the liquid to the soil is greater than the cohesion of the liquid to itself. So immersing a glass tube in a small diameter of glass tube of small diameter into a vessel containing water we can actually say that the rays of water in the tube is a function of the diameter of the tube. So here the diameter of the tube is nothing but the pore diameter and also the cleanliness of its inner surface that is something like we have got a soil solids which are not contaminated then that is where we can say that alpha is equal to zero for the cleanliness of the inner surface. So in the capillary rays what we said is that is a phenomenon where it can water can raise in the zone of completely saturation above the ground water table. So here this is the real path of the capillary path and at this point what you can see is that these are the surface tension forces which are actually acting in pulling the water against the gravity. So this is idealized here and shown here which is with the water table water pulling against the gravity. So it is reasonable to assume that the pore spaces between the soil particles of various diameter behaves in much similar to a capillary tube that is a tube of fine diameter. So if you look into this we can calculate the capillary suction. See here the capillary suction which is nothing but the suction force with which the water is pulled against the gravity. So here uc is nothing but force which is nothing but the surface tension force divided by the area which is area is nothing but the pore area. So here uc is equal to t which is nothing but the surface tension, pi d is nothing but the perimeter, d is the diameter of the pore and cos alpha which is nothing but the component I have taken here. So uc is nothing but t pi d cos alpha divided by pi d square by 4 this is nothing but the capillary or suction pressure. uc is nothing but now 4t cos alpha by d for clean tubes alpha is equal to 0 or the interconnected voids whatever we are seeing here if they are non-contaminated then we can take alpha is equal to 0. With that uc can be further simplified as 4t by d and with that we can write also 2t by r. So we can see here the smaller the pore radius then larger is the capillary suction. You can see here uc that is capillary suction is higher for fine grained soils. Now we can also use this further to calculate the capillary height that is nothing but the rise of the water against from the ground water table. So here the weight of the water is acting downwards. So weight of the water is nothing but pi d square by 4 into hc if hc is the zone of complete saturation pi d square by 4 hc into gamma w which is nothing but the weight force which is nothing but t into pi d I have taken as a clean tube with that alpha is equal to 0. So cos alpha component is 0 cos alpha component is equal to 1. So with that we get t into pi d is equal to pi d square by 4 into hc into gamma w with that we get hc is equal to 4t by d gamma w d is nothing but the pore diameter. So you can see here now this capillary rise also is appear to be higher for the soils which are actually having finer pores. So now in soils particularly the shape of the void spaces between the solid particles are unlike those in the capillary tubes. So the voids are irregular and varying shape and size and interconnected in all direction. This accurate prediction of the height of the capillary rise in soil is next to impossible but this can be estimated or can be also be modeled experimentally. However the features of capillary rise in tubes are applicable to soils as they facilitate an understanding of factors affecting the capillarity. So here the microscopic view of the soil is shown and here this is the groundwater table and this is the capillary rise. So the tube containing water exhibits positively positive capillary rise whereas the water adheres to the sides of the tube causing the fluid to rise slightly above. So this is already we have discussed the capillary pressure or capillary suction. So this is how it has been obtained like f is equal to t0 into t or t0, t0 is nothing but the surface tension of water and pi d cos alpha and uc is equal to f by a. So thus that is actually given as pi d t cos alpha by fd square and this is now uc is equal to 40 cos alpha by d and with alpha is equal to 0 it becomes to 40 by d. So capillarity and soil water energy if you look into this the soil water exists in small spaces in soil as film around the soil particles. The smaller pores can act as capillaries. A capillary is a very thin tube in which liquid can move against the force of gravity. So water is attracted to the glass tube by the adhesion. So a thin film flows up the soil side of the tube. So while cohesion drags more water along. So the capillary rise is discussed here which is nothing but the vertical equilibrium of at equilibrium we have equated the surface tension forces with the weight of water which is being lifted against the gravity. So at equilibrium hc is the maximum so that is actually nothing but which is given as 40d by w. Now we have actually seen the expression for capillary rise. Now we also have two issues one is that capillary velocity and other one is the time of capillary rise. This discussion can be obtained like this considering here in order to reduce the rate of capillary rise according to Landau et al 1967, Beikerman 1970. The mean velocity with the assumption that Poiseuille law is valid we can write v mean velocity which is nothing but the velocity with which the raise of water is taking place again above the groundwater table v is equal to r square delta p 8 ht mu w. So r is nothing but the pore radius ht is the height of the liquid or fluid lifted at any instant of time due to existing pressure difference delta p. So delta p is nothing but the pressure difference between the surface tension pressure it is the difference between the surface tension pressure and weight pressure. Weight pressure is nothing but the weight of water is column of water being lifted against the gravity. So mu w is nothing but the dynamic viscosity of the water or the fluid. So delta p is the difference between the pressure due to surface tension forces and pressure due to the weight of the fluid lifted at any instant of time. So ht is the height of fluid from the at any time t if suppose here if this is maximum hc and ht is nothing but the any time t the raise of water within this. So some soils the capillary rise rate of raise is very very slow. So in that case ht is nothing but the rate of raise of water above the ht is nothing but the height of raise of water at an instant of time t. So delta p is nothing but the difference between the pressure due to surface tension forces which is nothing but we have derived that is 2t by r or 4t by d for clean tubes and weight pressure that is pressure due to the weight of water which is nothing but rho wg into ht that is nothing but the and this is nothing but the weight of water in the column that is the weight of water in the column here divided by the pore area. Pore area is nothing but the d is the diameter so d is the diameter means here what we have is that pi d square by 4. So with that delta p is equal to we can actually say that is the pressure difference between the surface tension forces and the weight pressure due to the weight of water. So further we can deduce this expression using hc is equal to 2t by r we have substituted that hc as 2t by r and using v is equal to r square delta p into 8 ht mu w so substituting here and writing in case of 2t by r for surface tension forces so for delta b I am substituting here r square into rho wg into hc minus ht so this is nothing but the expression for delta p this is actually substituted for delta p in the derivation here by 8 mu w ht. Now by substituting v is equal to dht by dt and integrating now the time required to raise the continuous capillary zone hc is obtained like this by substituting v is equal to dht by dt and integrating the time t required to raise the continuous capillary zone hc is given as like this t is equal to 8 mu w by r square rho wg and hc into natural logarithm hc by hc minus ht minus ht. So here also you can see the smaller the pore size the larger the time which is actually takes for the capillary rise the rate of rise will be smaller so t is actually here 8 mu w by r square rho wg into hc into natural logarithm of hc by hc minus ht minus ht. So here we have the expression for the rate of the raise of the rate of raise of water due to capillarity. This further Terzaghi has given by assuming with the help of Darcy's law by assuming that Darcy's law is valid for the phenomena of capillarity in principle we knew that capillarity phenomenon is not a flow situation because if you consider this as the datum if you consider this as the datum and if you take this if you take here if this is the datum the pressure here is 0 and the elevation here is 0 so the total head is equal to 0 here. Now let us assume that from the datum the elevation of this point is say hc units and the pressure here is minus hc so pressure here total head is here is minus hc plus hc which is equal to 0 but so the capillary phenomenon is not a flow situation is a phenomenon which is actually resulting due to interaction between soil water and air interaction so but here the derivation which is actually obtained for the rate of time required for the raise of capillarity based on the assumption that Darcy's law is valid for capillarity which is given by t is equal to n it is nothing but the porosity hc by k into natural logarithm of hc by hc minus ht minus ht by hc so this is an expression for determining so here also in the previous expression whatever we have seen we notice that the lower the value of the permeability larger the time it takes for the raise of water. So here in this particular slide a demonstration which is actually given here that if you see here larger the pore spaces the raise of water will be less that is that is what we have discussed larger the pore space for example sandy soils the raise of water due to capillarity will be close to 0 so height of the capillary raise is a function of diameter of capillary so that is what actually shown here and one more interesting thing is that the pore diameter is approximated as one-fifth of the effective particle size that is the 20% of the particle size is approximated as so if you take a fine grained soil the pore spaces are very fine if you take a coarse grained soil if it is having a d10 is equal to 0.5 mm then it is around 0.1 mm. So if you look into the capillary raise what we have discussed is that hc is equal to 40 cos alpha by gamma wd so this is an approximation for soils if you take t is equal to 0.00074 kN per meter and gamma is equal to 9.81 kN per meter cube and alpha is equal to 0 with the d is equal to ed10 we can write this expression as 30 by ed10 when d10 in meters so this is nothing but the capillarity raise is given by c by ed10 and this estimate may be improved to allow for the effect of the grading grain shape characteristics such as irregularity and flakiness etc. So hc that is capillary complete zone of saturation is nothing but the c constant by ed10. Further the c is nothing but a constant which is 0.1 to 0.5 cm2 is a function of the grain shape and surface impurities and e is the void ratio. So capillarity action holds the soil water in small pores against the force of gravity the smaller the pores the greater is the movement can be. So here in this slide the capillary raise in soil for different soils is shown and here different effective particle sizes are shown and the here the capillarity heights are shown here so you can see that for soil which is actually having which is having less than 2 microns the clay which is actually having 0.002 mm size the height will be several meters capillary height of raise of water above the ground water table is several meters for silty soil is about 1.8 meters and for coarse gravel it can be just of 6 cm. So here this particular whatever we have discussed is shown through a slide here this is the saturated level and this is the complete capillary raise including the partial saturation you can see that as you can see for gravel and with increase in the size effective particle size there is a decrease in the height of the capillarity. And as we discussed here the relationship between the grain size of uniform quads powder and height of the capillary raise. So hc is greatest for the fine grained soils but the rate of raise is slow because of the low permeability from the discussion also we have seen that the is maximum for silts and very fine grain soil particles. And the capillarity particularly for coarse grained soils if you wanted to estimate then you have got a situation here for dry soil it will be 0 and here you have got a negative pore water pressure will be there because of the phenomenon of the ground water table with capillarity phenomenon. So this is how the capillarity raise in the stress profile is drawn but if you look into the stress profile for effective stress due to capillarity. So here you can see is that these particular negative water pressure reduces to 0 because of the ingress of the air at this zone. So here this is the ratio will be effective stress will be like this and then this particular variation will be like this. So this is particularly because of the end of the zone of the complete saturation and this is nothing but the capillary fringes which are actually called. So this is for total stress and this is for the native pore water pressure in the zone of the capillarity and then above this reduces to degree of saturation from here to here reduces to from 100% to 20%. So further you know many interesting phenomenon can be explained with the importance of the capillarity particularly like when you have got if you have got a phenomenon like near the beaches or we have got with the you got a soil which is with the less amount of moisture the capillarity phenomenon can be explained in length. So what we have understood in this particular lecture is that the concept of effective stress has been introduced and we also have deduced the expressions for the height of the capillary raise and the velocity and the time required for capillary raise then we have discussed that the fine grained soil which is actually having very very small force that time for the raise of the capillarity may take a long time. But if you have so this particular phenomenon can be advanced used particularly for construction particularly if you have got a sashi ground where there is if you are constructing embankment on a soft soil then if you prevent the to prevent the ingress of the water into the embankment fill material there is a possibility that we can actually use the coarse grained materials to as a cut off. So nowadays with the advent of new materials in civil engineering we also have materials like non-woven geosynthetics which also they can be used as capillary cut off layers.