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Video 1
Hello everyone. We have seen an important constitutive relationship; it is the soil-water characteristic curve, which is a relation between water content and suction or negative pore water pressure, within the water. We have seen how theoretically this can be deduced for idealized cases, and after that, we going introduce, another constitutive relationship called hydraulic conductivity function. Because, whenever there is a flow that takes place through partly saturated soils or unsaturated soils, what hydraulic conductivity we should use to simulate the flow through soils. Because, as the water content in the soil increases the hydraulic conductivity or the flow would be higher, and when the water content decreases, the flowrate would be lower. So, with what hydraulic conductivity we should simulate the flow conditions, is a question. So, I have considered two simpler cases, where you have a soil column, immersed in awater reservoir. Then, there is a flow that is taking place into the soil mass. So, the water content of the soil is also increasing, as a flow takes place into the soil, and there is a moisture content variation as we have seen earlier. So, there may be some change in the moisture content theta, this is theta s, and this may be theta r or 0, and this is height or z. Theta s, it would be up to the air entry value, and then beyond that, the volumetric water content decreases. And it reaches the initial value; initial water content within the soil sample, maybe at air-dried state. So, now flow takes place due to what? There is a flow that is taking place against the gravity here in this case. So, it is not the elevation hat that dictates; there is something else that dictates that is a chemical potential of the pore fluid; that chemical potentialwithin the soil system that dictates the flow. So, therefore, there is an upward flow that is taking place. So now, when the flow takes place, with what rate the water uptake takes place within the soil, is a question. So for that, we need to understand whether Darcy’slaw is valid or not. And similarly, there is another case where you have soil mass; this is the ground-level and this is a water table; there is a ponding that has taken place due to rainfall. So, then,there is a flow of water that takes place into the soil mass; there is infiltration that takes place. That is, in the direction of gravity only. In the direction of gravity, there is a flow that takes place. So, how to simulate these things? We have one flow equation. We will discuss this little later, when we discuss the flow through unsaturated soils; which is called Richards’ equation which is commonly used to simulate the flow through unsaturated soils, like the cases I have discussed here. So, if you observe the flow equation; in partly saturated soils there is a ∂by ∂z; z and t, these are independent variables; and you have something called k h term. Hydraulicconductivity, k is the hydraulic conductivity which is a functional form that depends on matric suction head. This is a matric suction head, which is a matric suction head; which is nothing but, u s u w, divided by gamma w. So, there is a matric suction head. And, zagain, an independent variable h m is the same suction head, and theta here, is a volumetric water content; the volume of water per total volume. So here, essentially, we require, a relationship, between h m versus theta, because, you have dependent variable here in as a h m; here you have theta. So, this could be expressed in terms of theta completely, if you have a relation between h m versus theta.This is your soil-water characteristic curve. This we have defined earlier. How the matric suction varies with volumetric water content is given by the soil-water characteristic curve. Here, there is another relationship, another constitutive relationship that is required tosolve this particular equation; which is called hydraulic conductivity function; conductivity function. So, here this is what I have written, soil hydraulic conductivity function, HCF. So, this is HCF and this is SWCC. These two constitutive relationships are fundamental in unsaturated soil flow. When the flow takes place through unsaturated soils, these two constitutive relationships are fundamentally important for understanding the flow behavior.So now let us try to understand the HCF, Hydraulic Conductivity Function. So, when there is a flow that takes place, we commonly use Darcy’s law, which states the flux, or the Darcy velocity; small q is proportional to i, that is the hydraulic gradient that is Darcy’s law. If there is an equation constant, we put k that is a hydraulicconductivity. Here, we put a negative sign, because the flow takes place from higher head to lower head. So, this is Darcy’s law.Now, the question is, whether Darcy’s law is valid or not. Because when I plot, a relation between q that is a flux; and i, hydraulic gradient, I should expect a unique relationship between these two. So, that means, the slope of this one is hydraulic conductivity. So, this is constant. This should not depend, or this should not vary with hydraulic gradient, or it should not vary with anything else. It only depends on the pore structure, pore geometry etcetera. However, this is independent. This is a material constant. Let us see what happens in the case of unsaturated soils. In unsaturated soils, if I plothydraulic gradient on the x-axis that is i, and flux, that is q, on the y-axis. If I maintain, if I conduct a test, hydraulic conductivity test in a soil column by maintaining one particular head, one particular head I have maintained; and this is the soil column hashed area. Here, I have I happened to maintain particular water content. So, this is possible. Generally, what we do in a hydraulic conductivity test is that the soil is completely saturated and after that only we conduct the test and obtain the case hydraulic conductivity; saturated hydraulic conductivity. But in this case, I want to maintain certain theta, which is less than theta s which is less than the saturated volumetric water content. That means it is in a partly saturatedcondition. Even after maintaining a certain head, I should be able to maintain certain volumetric water content. So, because in the unsaturated state, you have a two-phase system in the pore space you have water and you may have another medium that couldbe air, or it could be other immiscible fluids, such as oil or something. So, I can maintain certain water content by maintaining the other fluid content to be the same. So, that is possible, right? So, if that could be maintained then, what relationship I obtain between flux and i? Again, I get a unique relationship like this. This is where theta 4 is the volumetric water content I maintained. So, this is theta 4 is much less than thetas, right? So, now, this is a unique relationship again like Darcy’s law, where the flux is proportional to the hydraulic gradient. Now, I change my volumetric water content, or I slightly increase the volumetric water content by decreasing the content of other fluid. Could be oil and that content I reduced.So now, I maintained something called sigma 3. Sigma 3 is still less than theta s, but sigma 3 is so greater than theta 4. So, this volumetric water content theta 3 is larger than or higher than the theta 4, but less than theta s only. So, then what will happen? My flowrate will change; my flow rate will increase. And this is a flow rate I obtain. And again, this is another unique relationship. Similarly, I can keep on increasing my water content; theta 2 and theta 1, or theta 1 may be equals to theta s only. So, such a way conditionalso, I can bring in. So, then my flow is changing. For the same hydraulic gradient, I may have different flow rates or different flux values. My flow rate is changed. Or, I have a soil sample. I just allowed the soil sample to; initially, the soil sample is at the air-dried state. Then I maintained a certain water level. So, then I allowed the flow to take place through the soil. Then what will happen? As and when the water passes through it, the hydraulic gradient also decreases because the level decreases. So, slowly hydraulic gradient decreases, but my volumetric water content is increasing; and which causes the increase in the flux. So, I go on this path. So, either way, it could be, it could be possible. By either maintaining a constant hydraulic gradient, I can achieve different flux or different flow rates with different volumetric water contents. Or my flux or flow rate is, should, can change with time due to an increase in the moisturecontent. So, therefore, whether Darcy’s law is valid now? This is valid. This could be used; Darcy’s law still could be used. Darcy’s law is, minus k i. Is still could be used, provided k, this slope is k 1. This slope is k 2. This slope is k 3. This slope is, say k s. Because this is a theta s. Corresponding to theta s, the hydraulic conductivity is theta s. So, k itself is changing with the change in the moisture content. So, when the theta volumetric water content is very small, then k is smaller. As volumetric water content increases to the saturated water content, then it is it became k s; so unless until you assume a functional form for ‘k’, it is not possible to use Darcy’s law. So, Darcy’s law could be used, by defining k as, k as functional form. K is a function of theta. So, this functional form is called the hydraulic conductivity function. Hydraulic conductivity function is k as a function of theta, or it could be k as a function of h m matric suction head, or k as a function of suction, matric suction, or simply suction.Because theta is again dependent on suction; so we can define k of theta or k of h m. So, we just said that q should be written as minus k of theta, functional form. And i, i is what? Hydraulic gradient, so that equals, minus k theta. So, there is a dou. You know the one-dimensional case. This is simply, ∂H by ∂Z. ∂z or ∂x. So, this is a special variable. Now, what is H? This H, this is the total head. Generally, in saturated soils, the total head is defined as, the head due to elevation, or elevation head, plus pressure head, plus thekinetic head or velocity head. Usually velocity heads, velocity heads we ignore; because, they are negligible compared to other heads. The velocity of water through soil pores is negligible. So, we generally ignore it. So, now, when you construct the elevation head and pressure head; I will take one simple example, where you have soil, which is connected to a reservoir. I maintained a certainwater level. So, this is the water level. This is the elevation head. Now, flow is taking place through this. There is an event. So, flow is taking place. This is one experiment I have conducted. I can conduct another experiment, by maintaining different elevation head. So, the height has reduced. Now, in this particular case, the elevation head isreduced. Compared to this, the elevation head has reduced. So, if this happens to be clay soil; compacted clay soil, such as bentonites or something, you would observe that the elevation head does not make any difference in the flow rate through soils. So, the wetting front, if you observe within the soil mass you would see that it moves at the same rate as it moved in this particular case; first case. That isbecause, first of all in the saturated system itself, hydraulic conductivities of saturated soils in bentonites are very less which is less than maybe 10 power minus 9 meter per second; which is very very, small. So, in that particular case, the hydraulic gradients arenot important. So, diffusion dominates. So, in this particular case also, because this is unsaturated soil, the k values are much lessthan k s. The hydraulic conductivities are far less than saturated hydraulic conductivities. So, the head would not make any difference. So, even if you have a very small head like this the flow rate would be the same. So, then what is it that governs? The elevation head remains the same. The pressure head is not there. Pressure head, is not anything is maintained, so then the total head remains the same. So here, the elevation head is changed. And, the pressure head is not changing, or which is not there. So, when the total head is changing, but the flow rate remains the same, what could be the reason? When you are changing the elevation head, when you maintain at say 10 meters also, the flow rate will be the same. So, this is not increasing as high as this. So, then what is it that governs the flow here? The flow here governs, other than the elevation head etcetera. There are other things, like chemical potential. So, the total head now should consist of in unsaturated soils, the elevation head plus if you have pressure head, positive pressure head. But then generally, this is a negative pressure in unsaturated soils. So, this pressure head is a negative pressure head within the soil mass. Now, this one is a chemical potential head or matric suction head that can be represented as h m. Matric suction head plus if you have salts then, it could be the osmotic suction head. So, this is what governs the flow behavior. So, as a matric suction head in this 2 particular case is significantly larger compared to the elevation head you are talking about, this a negative value. This is a negative quantity, which is very large compared to the elevation head you are talking about. The matric suction head, as I have mentioned in earlier cases that it could be as high as several 1000 meters. So, if this is a several 1000 meters, negative 1000 meters, and this you are maintaining 1 meter, plus 1 meter plus 0.1 meter, if you are varying in this manner. So, it does not make a big difference on the flow rates. So, that is a reason why your flow rate remains constant, even though you are varying the positive head; theelevation head. So now, Darcy’s law, which is q equals to minus k, theta times, ∂H by ∂Z, this H should contain the total head. The total head should contain elevation head, plus matric suction head, plus osmotic suction head all the heads should be there. Then, this modified form of Darcy’s law could be used for flows. So now, we are not going into solving the flowequations and then understanding the flow through unsaturated soil. We will do it a little later. Now our interest is on hydraulic conductivity function.
Video 2
So, what is this hydraulic conductivity function? If I plot, the hydraulic conductivity versus volumetric water content on the x-axis, I would expect that the hydraulic conductivity increases as a volumetric water content increases; and it reaches a nearly constant value at theta s. Theta equals theta s. θs is saturated volumetric water content or this is (Refer Time: 20:07). So, at that particular point this is, this will become k s that is the saturated hydraulic conductivity. Or, this could be written, this could be plotted, in terms of matric suction. So, here its written relative hydraulic conductivity that is nothing but your k r is k by k s. So, when the soil is completely saturated, the hydraulic conductivity is k s. So therefore, k r is k r is 1. As the water content within the soil pores decrease; that means, only the contact angle is changing; the meniscus curvature is changing; small change in the water content. But definitely, the flow takes place through these larger pore space. So, therefore, hydraulic conductivity significantly does not change. So, until there is a, air entry that takes place. This is a, air entry point; so at this particular point, where air has entered into the soil. So, a sudden drop that occurs in the hydraulic conductivity, and with decrease in water content further, it significantly decreases. And it becomes close to 0 at residual state. At residual state, the water content exists as a thin film around the particles. So, in that particular case, the flow channels are not existing. Therefore, the flow does not take place. Vapor flow makes this. So, now we have seen that the, hydraulic conductivity function when we plotted in termsof k in terms of suction we have seen that the hydraulic conductivity initially changes;hydraulic conductivity initially changes slightly, and then after that it decreases significantly. So, after reaching the air entry, we have seen that, there is air enters into the larger pores or largest pores of the soil system. When the larger pores are getting emptied, air enters. The macro flow channels do not exist anymore; and flow has to takeplace through the micro channels; small thin channels. So, compared to the macro channels, the micro channels offer more resistance for the water movement. Therefore, the flow rate or the hydraulic conductivity decreases drastically. So, if you consider thin pipes; considering a thin, so many numbers of pipes, and considering one single pipe and the volume of voids in both these cases remainsame. We have considered so many numbers of pores here. N number of pores and only single pore here single larger pore and a small pores of N number volume of voids in both the cases remain same. So, in that particular situation, the flow through this larger pore will be highest compared to this particular case. Because, larger pores first of all, if you, from the Poiseuille’s equation if you consider, the volumetric flow rate, q is equals to pi r power 4, divided by 8 nu into change in the pressure by L. So, apart from the viscosity, and the length of the pipe etcetera, and change in the pressure; and this is a constant. Now, the flow rate isdirectly proportional to the r power 4. And the r is reduced to half, then Q will reduce to, 1 by 6th. So, flow rate will decrease one-sixteenth value. So, similarly here, because you have, so much of, here, it will have more resistance for the flow to take place, because, you have so much of surface area so much of surface area. There is a huge resistance that is offered from the wall surface. So therefore, here as soon as the larger pore get emptied, due to the air entry.S o, the hydraulic conductivity drastically decreases; and when the capillary action is taking place here, due to the reduction in the water in the larger capillaries, the hydraulic conductivity decreases drastically. When compared to volumetric water content versus suction, that is SWCC; that relationship may be somewhat like this. The hydraulic conductivity function k versus psichanges in this particular manner. So, this is air entry. This is air entry. Beyond that, there is a change in the volume, volume of water. The water content decreases. So, because, decrease in the water content especially this slope is much steeper compared to this one. Because, here as the larger pores are getting emptied, the hydraulic conductivity decreases very sharp. So, probably I can adjust this, by plotting like this; much steeper one. This is much steeper. So, once it reaches the residual portion, you would see that the hydraulic conductivity goes to nearly 0. Because, here, when you apply a huge suction the volumetric water content decreases in significant amount. So but then, that is a mechanism here. But here, the mechanism is that, when the water exists as a thin film around the particles, as we have seen earlier this is a particle, and this is another particle. So, water exists only around as a thin film around the particles. So, in that particular case, the pores are emptied, but flow cannot take place through these channels. There is no channel exists. Actually, channel broke down, because, the larger pores, and small pores are also got emptied. Now, you have water only availableas a fill. Therefore, the hydraulic conductivity goes to nearly 0; 10 power minus 16 or something. These are theoretical values. But, experimentally we cannot measure 10 power minus 16; 10 power minus 10 meter per second values, right? So, that is the hydraulic conductivity function.So, we require both these functions. SWCC and hydraulic conductivity function for understanding the flow behavior; so as SWCC is hysteresis, it has, this is a drying curve, and this is a wetting curve. Similarly, the hydraulic conductivity function also will follow hysteresis. So, this is the drying path and this is wetting path. So, drying hydraulic conductivity function lies above the wetting hydraulic conductivity function. However, we will see that because at a given suction the water content in the wetting curve is smaller than the drying curve; therefore, when the water content decreases, sodefinitely the hydraulic conductivity also decreases. So, this follows the same logic as SWCC. As the HCF, hydraulic conductivity function replicates the behavior of soil water characteristic curve, often the hydraulic conductivity function is derived SWCC. We will see how to derive this hydraulic conductivity functions for a given soil from soil water characteristic curve, in the later lectures. So other, there is a interesting concept that also we observe in hydraulic conductivityfunction is that, when you plot hydraulic conductivity versus suction for different soils. For hydraulic conductivity of sand and clay when you are plotting, definitely the hydraulic conductivity of sand is saturated hydraulic conductivity of sand is much higher, and saturated hydraulic conductivity of clay is much smaller to sand. This is sand. k s of sand, and this k s of clay. So, how do you plot? You have a small air entry; generally, sands will have very small air entry. Immediately, the air enters into the system. Because, when you take a small sand sample, saturated soil sample, with minimal effort, water comes out; under gravity itself. So, maintaining saturated condition in sand is very difficult. Unless and until, you have acontainer, and pour the sand, and put the water. In normal condition, when you hold it in hand, the water drains out under the gravity. And then, you generally have a unsaturated soil; that means, under the gravity itself, the soil exists as unsaturated state only. That isthe reason, why we usually make sand castles on beach side, because soil condition could be maintained unsaturated.So, here hydraulic conductivity of sand is very huge; but, air entry is very small. So, immediately as and when the air enters into the system, the hydraulic conductivity drops and then it follows, it follows like this. So, this is for sand. So, in case of clays, the hydraulic conductivity is very small; and the air entry value is very high;, very very high. Significantly higher than the sand and it follows, it follows this path and it goes in this manner. If I extend this curve, then it follows like this. Let us understand another important aspect in hydraulic conductivity function. That is, dependency on the type of soil. So, if I plot for 2 different soils, you would see that, for example for sand this is k and this is suction. For sand, the hydraulic conductivity is very high. So, the k s of sand is very high. So, I expect that hydraulic conductivity slightly decreases with increase in the suction, up to the air entry. Air entry of sand is very small compared to; air entry value of sand is very small compared to the clays or other soils, finer, fine grained soils. Air entry of sand is smaller compared to the other fine-grained soils. This is obvious, because when you hold a saturated sand in hand, water simply drains outunder the gravity itself, because, sand does not have the ability to retain water against the gravity. So, because the capillary size is larger, it does not hold the water that much. So, that could be understood using simple capillary mechanism, when you have twocapillaries; one is a larger capillary and another one is smaller capillary. The height of water in this capillary if you see the height is smaller in larger capillary like sand; this represents sand and this represents fine grained soil. So, fine grained soils will have larger air entry value. This is the air entry for sand, this is the air entry for fine grained soils.So, therefore, generally it is difficult to maintain full saturation in sand, because, under the gravity itself the water drains out. So, that is the reason why, when we make a castle, sand castles; obviously, the water drains out and then it maintains only the unsaturated state; it has more strength compared to either fully saturated or fully dried state. So, here the air entry of the sand is very small and beyond that you expect that the hydraulic conductivity drops drastically. And then it goes to very small value maybe 10 power minus 16 or 18. Or something meter per second very small value. Insignificant value, it becomes. In case of clays, if you consider, clays or any other fine-grained soils, the hydraulic conductivity is much smaller compared to sand. So therefore, the saturated hydraulic conductivity of clays lies here. The first point lies here. So, then the air entry value if you see, the air entry value is very high compared to sand. So therefore, for very large values of suctions, it retains water and after that it decreases the hydraulic conductivity. Here, in this case, it is much steeper than this particular curve. Because, as a larger pore get emptied, most of it is like, there is no flow channels and immediately it goes to nearly 0, or very small value it becomes. Fine grained soils, you have more number of smaller pores. So, more water is retained; and as and when they get emptied, slowly the hydraulic conductivity decreases. So, this is much steeper. In sand, this is much steeper. This is a very interesting observation because, interestingly beyond this particular suction, I name it as psi b beyond this particular suction clay or the fine-grained soil, this is fine grained soil. Fine grained soil will have hydraulic conductivity, this after, beyond this particular value. The finegrained soils will have hydraulic conductivity. If you consider any suction beyond this psi b, at any given suction, the fine-grained soils will have higher hydraulic conductivity compared to the sands.So, this is beyond psi b. This became vice versa, where fine grained soils will have a hydraulic conductivity much larger than the sands. So, sands will have lower hydraulic conductivity. So, this has important implications. Because, when you take a dry, fine grained soil, and when you take dry sand, this is sand. And this is fine grained soil, if you consider this setup. So, when the flow takes place through this; that means, initially they are dry. So, initially the suction is here[vocalized-noise], very high. The suction maybearound 10 power 6 kilo Pascal. So, this is very high value, air entry value, near air entrysuction. So, at this particular point, you are moving in this direction. You are moving in this direction. So, because this is wetting you are making it to wet. So, when the flow takes place through this; through fine grained soil when the flow takes place. So, suction in theclay is decreasing. So, this moves in this direction. And, when this moves in this direction, at any given suction value, when this approaches here, at that particular suction value; for example, at this value, this will be extended further this will be meeting atsomewhere here. So, hydraulic conductivity of sand is much smaller compared to hydraulic conductivity of clay. So therefore, here, sand will have, will have, much lower hydraulic conductivity compared to clay or fine-grained soil. So therefore, flow will be arrested here. So, there is a phenomena that is observed in the nature, in many geotechnical fields situations. So, this has lot of important implications. So, hydraulic conductivity of sand could be muchlower than hydraulic conductivity of fine grained soils under one particular suction value or under particular suction range.
Video 3
So, far we have seen the hydraulic conductivity function of so partly saturated soils andsoil water characteristic curve of partly saturated soils. Once we defined these twoconstitutive relationships, we need to understand how to determine the state variables.So, we have defined several state variables such as water content. It could be gravimetricwater current or volumetric water content. And volume of soil, that is also required inmany places, and suction; osmotic suction, matric suction. So, depending on that,whether you are estimating only the matric suction or total suction; total suctionincludes, matric suction plus, osmotic suction, and hydraulic conductivity or, hydraulicconductivity function.Hydraulic conductivity is not a state variable. I just mentioned that all the state variables.Hydraulic conductivity is a material constant, but estimation of hydraulic conductivity atdifferent states of soil. That is at, this is a functional form k of theta or k of psi also needto be estimated. How to estimate or measure these values in the laboratory and field?That we will see now. There is a important relationship for SWCC, soil watercharacteristic curve.
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