 Welcome to lecture series on advanced geotechnical engineering. We are in module 3 lecture number 11 on compressibility and consolidation. So in the previous lecture, we have discussed about Barron's theory of three dimensional consolidation and we attempted to solve some couple of problems involving sand drains. As been discussed that now the current trend of the vertical drains is in the era of free fabricated vertical drains. So we will try to discuss about the more about the characteristics of free fabricated vertical drains and then we will try to do some couple of problems which are with the free fabricated vertical drains. Then we will try to discuss about how these PVDs can efficiently be used in combination with vacuum consolidation which is the upcoming research topic in advanced geotechnical engineering frontier areas. So this is module 3 lecture 11 on compressibility and consolidation. So we are under the theme here methods for accelerating consolidation settlements and radial consolidation and this is the continuation of the previous lectures. So in the previous class, we have discussed about a problem. Now the solution will be discussed. So in this particular slide, a problem wherein a oil tank is to be constructed on a soft clay. So for this what has been done is that in a circular embankment has been constructed with a set of the sand drains spaced at certain square pattern. So we need to calculate the spacing of the square grid of the sand drains such that the tank is constructed the maximum primary consolidation settlement should not exceed 20 mm. So that means that what we have to do is that we have to calculate the final consolidation settlement and the final consolidation settlement minus this settlement that means that that much degree of consolidation up to that much degree of consolidation we have to have both vertical and radial to be considered. And then the given properties are that we have been given coefficient of volume compressibility and coefficient of consolidation vertical direction and in the horizontal direction and diameter of the well has been given and the desired degree of consolidation is also given for 90% in 6 months. So the time required to achieve this target consolidation both in vertical and radial consolidation is about 90%. So let us look into the solution. The solution runs like this. First we can calculate the so-called rho C the final consolidation settlement and this works out to be 151 mm wherein you are required to allow only about 20 mm settlement. So 151 minus 2, 151.2 minus 20 divided by 151.2 you will get the desired degree of consolidation but let us take UVR as 90% in 6 months. So for this UVR if it has to happen in 6 months and the thickness of the clay is 7 meters in a single drainage because the bottom is actually having stiff clay and TV is equal to 3.5 into 6 by 12 because 6 months divided by 12 converted into years and what you get is that this 3.5 that is constant of consolidation in vertical direction divided by 7 square so which is nothing but 0.0357. So by using TV is equal to pi by 4u by 100 whole square we can actually calculate UV is only 19%. Now by substituting u is equal to UVR is equal to 90% and UV is equal to 90% in Karelo 1942 equation wherein UR can be calculated. So solving for UR you get 88%. Now that means that whatever the arrangement we do with the so called the drain arrangement we have to achieve the consolidation in this order. So the TR is equal to TCVR by DE square. Now the DE square the DE is nothing but in square arrangement we have been told. Now N is equal to DE by 2RW or which is nothing but DE by DW is equal to RE by RW. Now here what we do is that by substituting this N is equal to RE by RW. So with that what we can do is that DE is equal to we can see that DE is equal to 2RE. So that is nothing but when you substitute this what we get is that TR is equal to 6.2 into 6 by 12 divided by 4N square into 0.3 by 2 whole square. So this 0.3 by 2 is nothing but the diameter of the well divided by 2 that is in terms of diameter of the well and N is nothing but the ratio between the influence diameter that is DE by DW. So with this what we have got is that TR expressed in terms of 34.4N square has been actually obtained. Now after having obtained now in terms of N square TR is equal to 34.4 by N square. So we have to do the trail and error method. So let us see that N is equal to 5 that means that N is equal to 5 indicates that the drains are placed very close and with that TR is equal to 34.4 divided by 5 square which works out to be 1.376 and which leads to 100% when you compute for N is equal to 5 you compute FN N square by N square minus 1 into natural logarithm of N minus 3N square minus 1 by 4 N square. Once you compute you will get FN and after having obtained the value of TR then you compute UR is equal to 1 minus e to the power of or exponential to the raise minus 8 TR by FN. So this works out to be 100% that means that the desired consolidation is only radial consolidation is 88% that means that we are on the conservative side. Now let us try another value N is equal to 10. So with the TR is equal to 34.4 divided by 10 square which reduces to 0.344 and now again by computing FN and UR and UR value the computed value works out to be 85% which is less than 88% this order of N is equal to 10 appears to be N is equal to 10 or N is equal to 9 appears to be in order. So we can consider even N is equal to 9 or N is equal to 10 if consider N is equal to 10 then what we can do is that the spacing can be determined for a square arrangement because we said that for square arrangement DE is equal to 1.13S so radius effective radius is nothing but divided by 1.13 by 2 so with that SE is equal to RE by 0.56 which is nothing but N into RW by 0.56 with that what we get is that RW is 0.3 by 2 so 10 into because N is equal to 10 that is what we selected if you select N is equal to 9 you will get 2.5 meters or 2.6 meter spacing so 10 into 0.3 by 2 divided by 0.56 so with that we get a spacing of 2.67 meter so what it says is that for the desired consolidation of 90% to happen in 6 months theoretically without considering the speed effects and other aspects it works out that the 3 N mm diameter vertical drain is required at 2.7 meter center to center or it is in the order of like 2.6 to 2.7 meter center to center distance. So in this particular problem what we have done is that we have calculated U R from the you know given target UVR value and then after having calculated U R we try to match this with you know with the different assumed N values by trade and error method N is equal to 5 N is equal to 10 so with that the one which is actually a layout or arrangement or configuration which gives the closer value to the desired U R value is actually selected and after having obtained so this implies that the settlement will be of the order of not more than 20 mm when the tank is actually constructed and consolidation happens. So that you know this limiting or tolerable settlement is decided by the engineering charge or the design engineer based on the requirement or conditions of the site and also the requirements are stipulated by the client. Now in the similar lines this problem can be solved at your end by a road embankment is constructed on top of a 9.2 meter thick layer of clay sand which would be between silty sand at the top and dense sand at the bottom that means that it is actually having double drainage and the required degree of consolidation before the embankment construction is 90% within 9 months that is with double drainage and with sand drains but that means that U V R vertical consolidation as well as radial consolidation has to happen within 9 months and for this purpose sand drains of 450 mm diameter need to be installed in a square arrangement. So you can also consider you know triangular layout and the square arrangement and you can compare you know how you know these you know spacings are actually varying. So estimate the spacing of the drain and from the laboratory test what has been obtained is that C H is equal to 0.288 meter square per month and C V is equal to 0.187 meter square per month. So in calculating TR the units need to be you know considered properly and here in this in addition to what has been actually asked we also calculate for the triangular arrangement also so that the layout of the sand drains can be compared in this problem wherein you know the procedure is actually very similar. So what we need to calculate is that you need to calculate you know what is the you know for 90% and then you need to calculate what is the vertical consolidation and afterwards you know you need to calculate U R and afterwards we need to calculate you know the TR time factor in terms of you know some constant divided by n square. So with that for different values of n is equal to 5, 6, 7, 10 you can actually get what is the you know value which is actually closer to the desired degree of consolidation in the radial direction and after obtaining that you know you can actually calculate based on the you know the spacing of the drain can be estimated you know for a given degree of consolidation which is satisfying the desired degree of consolidation and with that it can be used for compared with square arrangement and triangular arrangement. So before looking into the details of characteristics of PVDs we will try to look into some you know typical applications where you know suppose if you are having approach road embankments on soft clays and one of the viable options is to you know if you are the load or the you know capacity if the consolidation required to be accelerated you know during the construction of you know this approach embankment. So one of the alternatives is to go for you know the PVDs below the you know below the base of the reinforced soil wall and here a typical cross section of a back to back RU wall is actually shown wherein it actually has got a facing and a foundation pad and the drains are actually placed along with you know a drainage blanket at the top. So this is one of the applications for PVDs for approach road embankments and this is you know application of PVDs in ash pond dam risings like you know as you know we have the starter dyke which is actually constructed on the by using the natural soil and then the ash is actually filled up so this generally happens in you know three stages or four stages So in order to ensure you know the accelerated consolidation and also in nowadays if that area is in the you know earthquake prone area at the onset of you know say earthquake the liquefaction susceptibility can be you know mitigated by you know using you know this PVDs in ash pond dam risings. So it actually has got two advantages one is that it increases or accelerates the consolidation so once accelerate the consolidation means thus the fly ash or pond ash beneath the stage 1 embankment and subsequently stage 2 embankment and stage 3 embankment will get densified and the possibility of you know you know raise of excess pore water pressure at the onset of earthquake perturbance can be averted. So this is one of the you know very novel application wherein you know the PVDs actually is used for mitigating liquefaction susceptibility. In fact another application of PVDs is that you know in some areas where contamination flumes are there and in order to drain out the contamination training contaminating fluid this is one of the alternatives used to you know use PVDs and also nowadays in municipal solid waste landfills for a bioreactor landfills in order to went to the gas you know these you know use of PVDs is actually is coming out as a one of the excellent options and also collecting the gas uniformly. So with that what will happen is that the gas collections will be uniform and then you know the waste strength actually gaining will be uniform that you know they has got a crude benefits as far as you know bioreactor landfills the use of PVDs in bioreactor landfills is concerned is one of the novel application and in addition for the PVDs are also now being used along with the in silty sand and sandy soils basically to safeguard or you know again is the when the attenuation of liquefaction actually happens because of the sudden or catastrophic disturbance caused by earthquakes or due to some loadings like blast loadings and these are actually proved to be very very efficient and economical and lot of actual research is actually happening in these areas. Now let us look into the typical details of PVD so free fabricated vertical drain so in this particular you know slide what we see is a you know a typical plan view and cross section of a you know PVD and what we see is the filter jacket and inside is the core and between the cores or the channels and it is connected it is covered by a filter jacket. So as we said that this filter jacket is actually generally made with nonwoven Jotak style either with polypropylene or polyester or any other appropriate material and this core which is actually made with you know the HDPE or you know polypropylene core basically it actually has to be efficient in draining the water even under you know stress even under this conditions when the stress is actually subjected. So the rectangular cross section consisting of a synthetic Jotak style jacket surrounding a plastic core basically they are made with PET and PP Jotak style that is what we have just now discussed. Now each and every component actually has got a specific function so if you look into these functions of the drain jacket or filter jacket which is surrounding this plastic core is prevents the closure of the internal drain flow under lateral soil pressure that means that when the soil pressure is exerted in the lateral direction you know it prevents actually the closure of the internal drain flow under lateral soil condition and to form a surface which allows the natural soil filter and to develop to inhibit movement of soil particles into the drain. So it actually also helps to you know form a natural soil filter to develop you know or to inhibit movement of you know soil particles into the drain. So soil filter is actually developed to inhibit the movement of soil particles into the drain. So that is the you know the prime functions of the drain jacket and the other you know functions of the drain core are to support the filter fabric that means that it gives and provide the longitudinal flow paths along the drain length. So the plastic core you know functions are that you know it provides the longitudinal flow paths along the drain length and to provide the resistance to the longitudinal stretching as well as the buckling of the drain to provide resistance to longitudinal stretching as well as buckling of drain. So with the presence of you know the drain core it actually resists the you know longitudinal stretching as well as the buckling of the drain. When we look into the quality of this PVDs we will understand that these during the installations they have to sustain certain tensile stresses otherwise what will happen is that you know the possibility of 100 mm you know width reduces to about you know 80 mm or 70 mm because of the necking formation which actually takes place. So if the necking actually occurs you know very difficult to ensure you know the so called discharge capacities is one of the prime component which is actually is required in the design. So in this particular slide we have discussed about the two prime functions of a components one is drain jacket other one is drain core. Now this is the you know typical scheme of you know PVD installation we have seen the installation in the site in one of the sites in Navi Mumbai wherein this is a schematic actually being shown here. So the role is actually loaded and it is being driven with an anchor and you can see that you know the once it is in place here the mandrel is actually withdrawn so here there is an anchor is drawn. So these mandrels are actually having different ships they have got rectangular mandrel surrounding the drain or you know the rhombus shape you know mandrels basically these mandrels have been designed from the construction practice point of view to reduce you know the so called smearing effect or disturbance during the installation of these prefabricated vertical drains. So here you know a typical installation stage wise installation of you know PVD is actually shown schematically. So in this particular slide you know typical anchor bars which are actually used to keep the PVD in place during the installation. So in this particular photograph what you know has been shown is that how the anchored rod of Tullamum diameter and with a you know overlap of about 150 mm and here these staple pins are used to hold this in place to form a loop like this and then you know then you know this anchor bar which is actually having 200 mm that means that if 100 mm is the PVD breadth then twice the you know breadth is actually used and you can see that this is placed here and then you know this is the mandrel this is the rectangular mandrel what we are seeing is a rectangular mandrel. In this case this is with the anchor plate arrangement wherein the plate will be there and then you know this is actually there is a possibility that this anchor plate can cause more disturbance you know as it is being pushed into the soil and you know it actually has got the tendency of developing the you know large smear zones and this is actually schematically shown in the photograph here how the anchor plate is actually attached along with the PVD. So this is the anchor bar and this is the anchor plate. Now as we have discussed that you know these PVDs actually they come in the roles so we also have to take enough precautions to have the continuity in the sense that you know if we when we actually have you know roles which are attached so they should be continuity otherwise what will happen is that the flow channels will not be continuous and then there is a possibility that you know that discontinuity when it occurs at number of location drains will not be efficient. So in order to avoid these things for the typical type 1 and type 2 drain you know what the field practice actually says is that this is the existing running role and then you know the filter can be peeled out like this and then it can be inserted so minimum 300 mm lap and then these are actually placed again so that the soil will not enter and this is priced back here. So here in this first step open the filter backward for about 300 mm of the old drain as shown here then afterwards insert the opened core into the new drain to ensure the smooth overlap and fold back the filter secure the opening by tape and a staple actually as shown here. So you have to secure so that you know again soil particles will not enter through this particular zone because you know we have discussed that one of the functions here if the soil particles enter into the drainage channels the efficiency of the drains will get affected. And in the second type drain type 2 insert the new drain into the old drain and overlap the length should be again here also 300 mm and secure the opening by the tape and steps almost it is same it is universally it is adopted and also while testing the quality we also have to see how that joint is actually having effect on the tensile load strain behavior because if that war lapping is not adequate there is a possibility that the tensile stress and strain behavior will get affected in a way then is possibility that you know we develops failure and then discolourability. So because of that there is a possibility that you know the functionality of the desired functionality of the system may not be achieved. Now let us look into certain prefabricated vertical drain properties there are three basic things one is that equivalent diameter of drain so this was actually discussed in the previous lecture the conventional theory of consolidation with vertical drains assumes that the vertical drains are circular in cross section. So conventional theory of the consolidation with vertical drains basically assume that the vertical drains are circular in cross section therefore a band shaped drain need to be converted to an equivalent circular diameter. So a band shaped rectangular drain to be converted into a equivalent circular diameter which implies that equivalent diameter of circular drain has the same theoretical drainage capacity as the band shaped drain. So the conventional theory of the consolidation with vertical drains assume that the vertical drain so as discussed in the previous lecture the basic theories which are actually developed are for the circular cross sections. So now these drains these you know the number of equations and then parameter which are there have to be you know accounted for the different properties of the drains. So one of the you know important aspect is that tw is equal to 2 into a plus b pi that is for the equivalent diameter of the well so converting the rectangular area so equating the rectangular area which is you know pi dw is nothing but the perimeter of the perimeter of the circular area of the well equivalent circle is equal to the rectangular perimeter. So with that we get you know the radius as a plus b pi this is actually after Anspo 1981 and Rexner you know proposed that d is equal to a plus b by 2 dw d or dw is equal to a plus b by 2 and this was proposed based on some ferrant element analysis results. But however the universally this is actually adopted dw is equal to 2 into a plus b by 5. Now the filter and opening size as we said that you know the jacket filter jacket is actually made with nonwoven geotextile having certain apparent opening size and having certain mass per unit area. So the general guidance for the drain permeability is that k filter that is the permeability of the filter jacket has to be at least 2 times the permeability of the soil. So we know that you know most of the clays you know which are actually undergoing consolidation may permeability let us say 10 to the power of minus 9 meter per second minimum it actually should have 2 into 10 to the power of minus 9 meter per second and filtration requirement is given by you know o 95 by d 85 should be less than or equal to 3. So o 95 that is the you know opening size indicates that in approximate largest particle that would effectively pass through the filter and d 85 indicates the diameter of the clay particles corresponding to 85% passing. So o 95 by d 85 less than or equal to 3 and the retention ability of the filter is given by o 50 by d 50 less than or equal to 24 and the filter material can also become clogged if the soil particle becomes strapped when the filter fabric structure. So in this case o 95 divided by d 15 should be greater than or equal to 3. So clogging can be prevented if o 95 is greater than or equal to 3 times the d 15. So 3 to 4 times d 15 if it actually is there then the clogging can be prevented. So filter material of a filter jacket can become clogged if the soil particles become trapped within the filter fabric structure that is the pores which are actually there when they are actually occupied with the soil particles the clogging can actually get you know mobilized and so the clogging is prevented by ensuring that o 95 by d 15 is greater than or equal to 3. So the general guideline for the drain permeability is that the drain should have the filter should have more permeability than the soil and the filtration requirement is given by o 95 by d 85 less than or equal to 3 where o 95 indicates the approximate largest particle size that would effectively pass through the filter and d 85 indicates the diameter of the clay particles corresponding to 85% passing. So the retention ability of a filter is given by o 50 by d 50 which is less than or equal to 24. Similarly now another important parameter as far as the discharge capacity is concerned. So discharge capacity is basically is the flow volume of the flow happening through the PVD under a unit hydraulic gradient. So the discharge capacity of the PVD is required to analyze the drain or well resistance factor. Suppose the lower the discharge capacity of the PVD then you know then efficient is the you know consolidation of a consolidation effect on the consolidation performance of a PVD stabilizes soil. So in practice actually well resistance can develop due to deterioration of the drain filter may lead to significant reduction of the cross section and fine soil particles may pass through the filter and decrease the area available for the flow. So that means that the channels get blocked by the soil particles with that what will happen the effective area for the water to flow will get decreased and folding of the drain due to soil settlement may result in a reduced discharge capacity. So this we may have to note that the PVDs which are actually flexible in nature and when the soil is actually subjected to you know settlements and these PVDs due to two reasons one is that the soil is subjected to settlements because of the onset of consolidation and the second thing is that due to you know the presence of the lateral pressure the drains actually subjected to so called buckling that means that they can undergo at buckling at one location two locations and three locations. So even under those conditions you know in the buckling conditions it should be able to you know have the satisfy the required discharge capacity requirements. So the discharge capacity which is nothing but function of volume of the core and lateral earth pressure acting on drains. So this lateral earth pressure if wanted to be estimated this is actually obtained nothing but kh is equal to k0 sigma v at a given point if you are having a vertical effective stress and k0 you know based on normally consult soil if you are looking to eat 1 – sin 5. So we are able to take it as let us say you know 0.5 times sigma v you know is the lateral earth pressure at any point. So the possible folding which actually can arise due to the reasons which have been explained and the twisting of drain due to large settlement and you know it can also happen buckling or it can also undergo so called twisting and infiltration of the fine soil particles through the filter and the biological and the chemical degradation. So the discharge capacity of a PVD which is actually installed in the field can actually get affected by the following factors volume of the core lateral earth pressure acting on drain and possible folding and bending and twisting of drain due to large settlements and infiltration of fine soil particles through the filter and the biological and chemical degradation. Now this discharge capacity is given by qw is equal to ft into you know fc into ffc into q required. So these where fc and ft and fc and ffc are the influence factors due to time drain deformation and a clogging respectively. So these factors are that ft fc and ffc you know multiplied by q required. The term q required is the theoretical discharge capacity calculated from the Barron's theory of consolidation. So that q required can be obtained by epsilon f into u10 into L by into phi ch divided by 4th. So where epsilon f is the final settlement of the soft soil equivalent to 25% of the length of the drain installed to the soft ground and u10 is the 10% degree of consolidation, L is the depth of vertical drain and ch is the horizontal coefficient of consolidation and th is the time factor for the radial consolidation. So the actual discharge capacity qw is given by ft fc into ffc into q required and the term q required is the theoretical discharge capacity calculated from Barron's theory of consolidation with that q required is equal to epsilon f into u10 into L into pi ch divided by 4th where epsilon h is the final settlement of the soft soil equivalent to 25% of the length of the drain installed in the soft ground and u10 is the 10% degree of consolidation, L is the depth of vertical drain, ch is the urgent coefficient of consolidation and th is the time factor for radial consolidation. So if you look into this, according to after considering all the worst conditions like due to settlement buckling and then deformation of the drain due to lateral pressure that may occur in the field, the discharge capacity qw of the PVD could be around 500 to 800 meter cube per year. So most of the manufacturer's specified values are actually with 500 to 800 meter cube per year even under the lateral pressure conditions but reduce it to 100 to 300 meter cube per year where i is equal to 1 under elevated lateral pressure so this is actually after so most of the PVDs which are commercially available and they actually have the discharge capacity more than 100 meter cube per year. So most of the PVDs which are actually available have got the discharge capacity more than 100 meter cube per year. So if this is there then we can actually consider and if this qw is low then there is a possibility that it can actually get affected the drainage capability of the PVD which has been selected and used for draining water from the soil at the onset of consolidation. So in this particular slide a typical summary of discharge capacity is specified in the different projects actually shown and discharge capacity qw is defined as the volume of water per unit time that can conduct along the core of PVD in actual direction under unit hydraulic gradient. So qw is the discharge capacity is given by q by I and I is the hydraulic gradient and here you can see that the straight conditions and buckled conditions are actually prescribed here and in the Netherlands for the length of the drain less than 10 meters or the thickness of the clay layer less than 10 meters in straight condition it is greater than 10 and with this at the test condition is about 350 kilo Pascal's in 30 days and in case of buckled condition it is 7.5 which is when it is greater than 10 meters the straight condition actually should have k of 50 meter cube per actually second into 10 to the power of minus 6. So if you multiply this one with 31.536 we actually get into meter cube per year. So you can see that even under the buckled condition for this case here which is 7 into 30 is about 200 meter cube per year at 350 kilo Pascal's of pressure. So even under the buckled condition for the majority of PVD's they actually possess the discharge capacity in the range of 100 to 200 or 300 meter cube per year. So in this particular slide required drainage capacity as a function of PVD length and k of the soil is can be seen as the length of the drain is actually increasing and k of the soil is say decreasing then it can says that the discharge capacity falls down can be seen that as the length increases definitely yes there is an increase in the discharge capacity but let us say that for a length of about 20 meters this is the discharge capacity. So with increase in you know permeability of the soil then the discharge capacity requirement also increases because the larger the permeability the more is the water is pumped into the larger the permeability of the soil the more is the water pumped into the flow channels. And so the selected PVD has to be economical in understanding the requirement. So if you look into the liquefaction mitigation issues with the you know with the sand or city sand constant. The first of all the installation in these sand conditions sandy soil conditions a little bit can cause a worry and because of the frictional resistance offered to the mandrel and all, mandrel of a you know the tool which is used for installing these PVD's and then you can say that with the decrease in permeability you know that means that there is an increase in the required target desired discharge capacity for a given length. Now as has been told that you know because of the you know this jacket is actually supported to the core at number of locations. So the certain portion which actually will remain unsupported. So for a particular type of plastic core and which is like this and because of the soil it actually causes you know deformations. So you can see that the certain areas actually where the necking actually takes place and in addition to that if these elongation conditions are actually clubbed with this thing and then there is a possibility that it will get affected. So the reduction of the flow area caused by the deformation of the filter can be seen from this slide. So these conditions you know hamper you know the flow conditions through the flow channels and which in a way affect the performance of a PVD installed site. So this is you know a typical drain with and without smear zone has been told that this is a condition where if you are actually having a perfect drain and the smearing ratio that is ds by dw is equal to 1. If that is the case that you know then that s is equal to 1 wherein that is called as a perfect drain and when you are having you know this smear effect because of the disturbance caused by the anchor plate or you know the mandrel which is actually pushed along with the drain in the installation process as been shown before as it has been shown before it can actually cause the smear and this smear zone is actually referred here in as a disturbance, disturbed soil. So the lot of actually work is actually happening you know to define the smear zone as well as it was discussed in the previous lecture. These smear zone also actually has you know zones like transition zone and you know transition zone and then smear zone completely smear zone is actually close to the drain and then there is a transition zone where the soil properties you know tending to close to close to go close towards the undisturbed soil conditions and then there is undisturbed soil or undisturbed clay will be there. So the time required for the radial consolidation for PV drains this is actually equation is modified by Hansbo 1981 considering the you know disturbance effects and drain resistance factors. So Hansbo 1981 modify for the time the time required to achieve you know the radial consolidation or horizontal consolidation t is equal to you know time required for it to achieve uh is given by t is equal to dc square by 8ch you know within brackets function n and plus function s plus function r of the bracket close into natural logarithm by into square brackets 1 by 1 minus uh then the square brackets close. So here if you look into it there are three factors one is fn that we are aware of that and wherein we said that for drains without any smear and which is nothing but f function n is equal to n square by n square minus 1 natural logarithm n minus 3n square minus 1 by 4n square and for as you know n is equal to you know dE by dw and because you know this n value will be always more than 20 for PVDs. So 1 by n square is you know close to 0 and n square by n square minus 1 you will tend to become 1 so because of that you know this fn is actually simplified as ln to the natural logarithm of n minus 3 by 4. So fn is actually given by ln of n minus 3 by 4 or 0.75. So fs is nothing but the factor for the soil disturbance and wherein that this because of the smear and which is actually taken care like kh by ks minus 1 into natural logarithm of ds by dw. So ds by dw is also called as the smearing ratio that is diameter of the smear zone divided by the you know diameter of the well where s is equal to a smearing ratio s dash we can say and fr is the factor for the drain resistance. So this is because of the you know the possible effects ill effects due to the effecting of the drainage discharge capacity or drains discharge capacity. So the factor for the drain resistance is actually accounted which is fr is equal to fz into l minus z into kh by qw where in this particular expression for fr is equal to pi z into l minus z into kh by lw where z is the distance below the top surface of the compressible layer and then l is the effective drain length h for one way drainage and if h is the thickness of the clay layer and h for one way drainage and h by 2 for the two way drainage. In fact this you know when we try to simplify further when you take for the entire length then you get the expressions for the drain resistance factors fr for one way drainage and 2 way drainage within the drain. So by looking into this we can actually calculate the time required to achieve u h and if you are actually having this you know fr the f n divided by fr which is actually say you know something like less than 0.05 then you know the well resistance is actually can be ignored and you know it actually gone by the drain spacing only. Now after having discussed you know the PVD characteristics and you know the installation issues let us look into some couple of problems where PVDs were installed in a compressible clay layer of 10 meter thickness in a square pattern with a spacing of 2 meters and the PVD used is 100 mm wide and 4 mm thick and the coefficient of consolidation of clay is in the vertical as in horizontal direction and which is actually given as 2 meter square 2 meter square per year and 3 meter square per year and the boundary below the clay was impervious and what we need to calculate is the calculate the degree of the consolidation achieved in one year one year's time. So here you know the dimensions of PVD is given and C h and C v are given and the and here is also assumed that no smear and no well resistance you know need to be accounted. The solution works out like this first estimate d e because the layout which is actually given here is you know the spacing of 2 meter square pattern. So we can actually calculate d e is equal to 1.128 or 1.13 s into s is equal to 2 meters so with that we get the effective diameter as 2.256 meters which is nothing but 2256 mm and d w is nothing but equivalent diameter of the well which we have considered 2 into a plus b by pi so which is nothing but 2 into 100 plus 4 by 3.1415 which gives 66 mm then you calculate f n with a simplified version of ln n minus 0.75 so ln 2256 by 66 minus 0.75 which works out to be 2.78. Now the calculate the time factor due to radial drainage which is nothing but TCH by d e square where T we want in one year's time C h coefficient of consolidation horizontal direction is 3 divided by 2.256. 2.256 is nothing but effective diameter in the unit cell then you know it is 0.589 we get. Now calculate the degree of consolidation due to radial drainage where in 1 minus e to power of minus 8.589 divided by 2.78 after simplification we get 8 to 2 percent. Now the time factor for the vertical consolidation is about you know T C v by h square because it is one way drainage so 2 into 1 divided that is C v is 2 and 2 1 is the time divided by 10 square because of the single drainage is 0.02. So by using T v is equal to pi by 4 u by 100 whole square calculate you know the u v and using the Karelos equation we can actually calculate what is u v r. So by substituting u v is equal to 16 percent and u v is u r is equal to 82 percent and we can see that u v r is about 85 percent. So the degree of consolidation which can be achieved in 1 year time by inserting drains at 2 meter spacing of the particular PVD which is selected it works out to be of this order. Now let us consider the same problem assuming that smearing ratio is 3 that means that this particular performance is actually affected because of the smearing ratio and the coefficient of consolidation of the smearing soil is given as 1.5 meter square per year. So calculate the degree of consolidation achieved in 1 year time. So here for this FSN you know here it changes because wherein it actually has got the smear effect and with that what you actually get is that you know the so called you know FSN which by taking into account of the smear effect. So the degree of consolidation due to radial drainage is actually now is instead of you know in the previous problem we have got like 82 percent. Now because of the inefficiency of the drain because of the occurrence of the smear at the onset of installation the degree of consolidation radial drainage works out to be 70 percent. So again by using that vertical consolidation that is about what we have got in the previous problem as 16 percent we can calculate now with this you know the UVR works out to be you know UVR works out to be you know this value is not correct but UVR is actually reduced. So comparing the answers it can be seen that the degree of consolidation has reduced due to smear effect because now you know when once you take you know Carillo's equation by substituting this you can actually calculate the correct UVR value. So comparing the answers it can be seen that the degree of consolidation UVR has reduced due to smear effect. Now for this for testing these drains under straight condition or deformed conditions several you know testing methods actually have been device and this is a particular arrangement you know the drain with you know a deformed condition it can be seen that the flow is tested when the drain is under this type of deformed condition. So when we compare you know the discharge capacities of the drains with the straight and buckled condition you can see that the buckled condition actually exhibits you know buckled condition actually exhibits the lower discharge capacities. So here you can see that under the pressure so what we can see that you know 150 kilo Pascal's of pressure when it is applied the discharge capacity is around 16 to 10 to the power of minus 6 meter square meter cube per second whereas this was actually is about 80. So that means that there is a reduction of the discharge capacity because of these straight and buckled conditions. So we have to ensure that these discharge capacities are actually adequate for suiting to the soil conditions so all these conditions need to be considered. Similarly this is the tensile strength of the test for the PVD so it should have note that the tensile strength should be such that it can sustain the tensile load applied during installation. So minimum 1 kilo Newton is required at 10 percent strain and necking actually reduces the discharge capacity so necking should be avoided. So this is the typical you know constant rate elongation method testing for the you know combined PVD. So this is the typical tensile stress strain measured for the two typical drains you can see that the drain 1 and drain 2 and these are the typical specifications of the PVDs which are followed in China for L less than 15 meters, L less than 25 meters and L less than 35 meters and where in the thicknesses 3.5, 4 mm and 4.5 mm and you can see that the discharge capacities are about 670 and 1000 and say 1800 meter cube per year under the pressure of 300 kilo Pascal's and the tensile strength are actually that is kilo Newton per 10 centimeters is greater than 1 and 1.3 and tensile strength of the filter is also very important and tensile strength of the filter in wet condition is also shown here. So tensile strength of the filter in wet condition is actually have to be less than you know this one but in the previous curve actually it was noticed that the tensile strength of the filter in the dry in wet condition actually more than dry condition. So this is the circular type drain used for vacuum loading you know projects. So you can see that this is the typical drain used for vacuum loading projects which is you know specially when the drain is actually connected. So this is the schematic arrangement of the vacuum proof learning system now the currently the lot of research is actually happening in connecting PVDs directly to the vacuum pumps and with that what will happen is that the consolidation can be accelerated efficiently and without any preloading. So this is a schematic sketch is actually shown here where we have PVDs and then different components which actually has you know which is connected to vacuum pumps is actually shown here. So here you can see that the PVDs are connected directly to the vacuum pump so that the water is actually suckered into this. So here in this particular arrangement of when we combine PVDs with the you know PVDs in this case it is actually the PVDs actually have to do in case of vacuum preloading they have to do drain the water as well as apply the suction uniformly. So in this way you know this will facilitate for the better end efficient system. So in this particular module we actually have discussed about the consolidation theories and thereafter we actually have discussed you know the different types of conditions like normally consolidated and over consolidated soils and then finally we have discussed about methods for accelerating consolidation settlements, consolidation and particularly we have discussed in depth about the you know vertical drains you know usage in accelerating consolidation and as a forward path we can see that lot of potential is there for research in areas of with vacuum consolidation in combination with PVD and different applications of PVDs like as I said in the bioreactor landfills or mitigating liquefaction can be explored.