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Hello everyone, we were looking at the estimation of hydraulic conductivity function by multi-step outflow technique. We have also discussed the limitations of the technique and as well as the usefulness of the technique. The technique is useful for the estimation of both soil-water characteristic curve and hydraulic conductivity function to a maximum suction value of 1500 kilo Pascal if you use a 15 bar high air entry porous disk. There are other various techniques people often use on and off such as steady-state flowtechniques. So, we often use steady-state flows and transient flows through saturated porous media for the estimation of saturated hydraulic conductivity. Similar techniques are also used in the literature for the estimation of unsaturated hydraulic conductivity atvarious suction values by controlling suction. So, you may have a soil column, it is connected to two different reservoirs. So, now herewe maintain a certain head, and we allow the water to take place to go through it. However, in the unsaturated hydraulic conductivity technique, the airflow is alsoconsidered and high air entry porous disks are used at the top and bottom of the soil column. And here also similar to the axis translation technique or multi-step outflowtechnique, the air pressure is controlled, and a certain value of water pressure is maintained, and suction is controlled in the soil. And under this controlled suction, the flow is allowed to take place. Such techniques, we understand that such techniques are applicable for coarse-grained soils. So, in coarsegrained soils, it is easy to control the air pressure and maintain the water flow to take place through the soil. Otherwise, if it is a clay soil, you will we will not get anappreciable amount of water through the soils at the outlet. Similarly, often centrifuge techniques are also used where you have a soil sample placed in the centrifuge, and which is rotated at certain angular velocity. So, when it is rotated, a centrifugal force is applied to the soil mass. So, then the water flow will be faster. Compared to the saturated flow unsaturated flow through unsaturated soils, the flow rate will be very less. So, here in a centrifuge the flow rates can be improved or increasing the acceleration. So, such stress is also often used in the literature. However, such techniques are also applicable for coarse-grained soils, because in fine-grained soils, the soil will start settling and consolidation takes place. Therefore, such techniques are limited to coarse-grained soils only. So, thereforehydraulic conductivity function estimation for all soils may not be possible, and even for coarse-grained soils, the determination in the laboratory is very expensive. So, therefore often the hydraulic conductivity is determined from the soil-water characteristic curveitself. Once the soil-water characteristic curve is properly determined in the laboratory by using different techniques, using different ways that we will discuss very soon, the HCFs can be predicted, so such prediction models are very commonly used in the flow-throughunsaturated soils. Let us look into several SWCC and HCF models available. We also required to model the soil-water characteristic curve data, so that is when we get volumetric water content versus suction or so the θ versus ψ or gravimetric water content versus ψ or degree of saturation versus ψ. This data when we obtain in the laboratory such as these data points. We need to obtain a smooth curve such as this curve for the modeling purpose because of the soil-water characteristic curve and hydraulic conductivity function or required input data for the flow through unsaturated soils.So, therefore we require a smooth functional form between ψ and θ, and k versus ψ. So, these functions are often required for the hydraulic, for the flow through unsaturated soils. So, therefore we required to model the measured data from the laboratory are in thefield. So, we have several SWCC models available in the literature of unsaturated soil mechanics, such as a Brooks Corey model, which is proposed in 1964. So, this is θ equals to θ s when ψ is less than or equal to ψ b. Ψ b is the babbling pressure. So, this is a bubbling pressure, which is nothing but the air entry value, the air entry suction. So, when it is air entry suction or less than air entry suction, θ equals θ s. When the cross the suction crosses bubbling pressure, then you have θ r plus θ minus θ s minus θ r times ψ b by ψ power λ. Here λ is a pore size distribution factor and ψ b is the bubbling pressure or air entry value. So, in this equation, the λ values generally vary between zero-point small values maybe 0.1 or something. And it can go to very large values like 2 or so. Larger values of λ signify uniform pore size distribution and smaller values of λ signify the well-gradation. So, therefore this equation also can be written as, the bottom one can be written as θ minus θ r by θ s minus θ r equals to ψ b by ψ power λ, so which is nothing but which is referred with the bigger θ, which is called normalized volumetric water content. So, here θ r is the residual water content residual volumetric water content. And θ s is saturated volumetric water content. So, therefore when we plot a big θ that is normalized volumetric water content, and on the x-axis, you have ψ suction then for different values of air entry value. And for another air entry, so this is how it varies. The normalized volumetric water content varies from 0 to 1 because when θ approaches saturated value, saturated volumetric watercontent that equals to porosity. Then this whole thing will become 1, so there is 1. So, as a when the bubbling pressure is when the suction is less than or equal to the air entry value or the bubbling pressure, θ equals to θ s, therefore that equals to 1. And if ψ or suction is more than the bubbling pressure, then the normalized volumetric water content decreases from one, so this is how it varies. These three are for different air entry values for different AEVs. Similarly, for different λ values the normalized volumetric water content verses ψ varies in this manner. The AEV remains the same, so this is how it varies. So, AEV remains the same, but λ values are different.This is this maybe for λ equals to 0.5, and this may be λ equals to this is for λ equals to 2. So, λ equals to 2 are very large values indicate uniform pore size distribution. When you have uniform pore size distribution, you have a steep soil-water characteristic curve. And the residual water content is achieved at a very small value of suction because all the pores are uniform. So, water retention is not that significant, because it immediatelyloses its water as a suction increases. On the other hand, when you have well-graded soil, you have several different types of pores. You will generally have smaller pores in that particular case. So, therefore it extends, the residual water content is expected to be existing at very high values of suction. So, therefore the slope is not steep, and it extends to very large values of suction. So, this can be simulated very well using this particular expression given by Brooks and Corey, and which is the simplest possible model, because it has only one parameter to estimate. So, only the λ needs to be estimated, while fitting the data. So, the bubbling pressure can be usually when you plot in terms of normalized volumetric water content and suction, generally, the bubbling pressure can be identified or observed generally that is known. So, therefore only one fitting parameter is available. Often we may not know the residual water content value.And therefore, in that particular case, the two parameters need to be estimated from the data. So, this is a very simple technique that is the advantage of this particular model. However, the major disadvantage is that the discontinuity at air entry value, which has adiscontinuity. It reaches 1 up to the air entry vale and beyond that, it decreases, here there is a discontinuity. If you take the slope dθ by dψ, then this value is not defined at this particular point. So, therefore you cannot obtain a smooth function using BrooksCorey method.
And we have another method called van Genuchten model, which is very popular for the estimation of the soil-water characteristic curve from the measured data of θ verses ψ. So, the expression is θ equals to θ r residual water content plus θ s minus θ r similar to the Brooks Corey model times 1 over 1 plus alpha h whole power n, so here n should be more than 1. So, here this mod indicates the h should be positive value should be substituted here or alpha h should be positive. And because the expression is in terms of matrix suction head and matrix suction head is a negative value, so you will get complex numbers, if n is a real number. So, this is normalized volumetric water content equals to 1 by 1 plus alpha, often this iswritten as ψ by a whole power n, and this whole power m. Often this is written in this particular form in our geotechnical engineering. This initial expression was available in the soil science literature. When it is used in geotechnical engineering, we are acquainted or we often use soil water potential or suction directly. Therefore, the expression is modified into this, here ‘a’ represents air entry value, which is 1 over alpha when you compare with this expression. And ‘a’ is related to air entry value, but it is not equal to the air entry value. So, therefore, this is also a fitting parameter. Often it is shown on several soils and using the modeling also that a is not equal to the air entry value, which is related to air entry value. When you have a large air entry value and ‘a’ value is higher. When you have asmaller air entry value for a given soil, even a is small. So, qualitatively these two can be related, however there they are not quantitatively equal. So, the other parameters are m and n. So, n is related to the pore size distribution of the soil, m controls overall symmetry of the soil-water characteristic curve. So, when θ r is also not known, then you have four parameters to estimate or determine. So, we will see how generally these parameters are determined from the laboratory estimated data of verses ψ. Θ s generally known because when you conduct a test initially at the slurry state or whatever the state the soil is in a fully saturated state the porosity of the soil is known. Knowing the density of the soil, and water content one can estimate the dry density of soil from that one can estimate the void ratio. When the void ratio is known that is, it can be related to the porosity. And porosity at a fully saturated state is the volumetric water content at saturated or saturated volumetric water content, so that is how one can estimate the θ s in the test. However, θ r estimation is a little difficult. And sometimes, you can use θ r equals to as small as possible for clays, say 0.01 or something that people often use. And here, the ‘n’ value should always be more than 1. So, this is a constraint coming from the equation. So, let us try to understand, why n should be more than 1. If I write in terms of alpha h only and ‘h’ is used in positive terms, then this is the expression that I can derive. So, the normalized volumetric water content can be written as 1 plus alpha h power n whole power minus m. When you differentiate this expression dh, which is a minus m times 1 plus alpha h whole power n power minus m minus 1, therefore I can write it as minus m plus 1 and times n alpha h power n minus 1-time alpha, which can be written as minus alpha m n times 1 plus alpha h power n whole power minus m plus 1-time alpha h power n minus 1.If you see this expression as h approaches 0, and if you consider, n value to be equals to 1. So, this slope dθ by dh would approach minus infinity. So, the slope of the soil-water characteristic curve, if this is a soil-water characteristic curve. If this a soil-watercharacteristic curve at this point, where ‘h’ is very close to 0. So, the slope of the equation approaches minus infinity, so that means, this will never go to 0, but when it approaches minus infinity, so because of which what will happen is the diffusivity.Diffusivity is defined as K times dh by d θ approaches 0, so which is not physically correct. The diffusivity becomes 0, as K approaches Ks, which approached Ks, which is not equal to 0. So, therefore diffusivity should become 0. So, this happens, because when you consider n equals to 1, and similarly when n equals to less than 1 also when h approaches 0. So, this whole expression approaches minus infinity. So, therefore diffusivity approaches 0. If n is greater than 1, then this whole expression e is well defined, therefore generally the n is restricted to be more than 1 in this particular model. So, the van Genuchten model is very well received and the work by van Genuchten in 1980. So, this particular work or journal paper received more than 20000 Google Scholarcitations. So, this is this work is very very well received. And even till-date, the van Genuchten model is widely used for you know representing the soil-water characteristic curve data. And often used in the modeling of unsaturated flows. The major advantage of the van Genuchten model, when compared with Brooks Corey's model is, it can provide a smooth SWCC curve. So, there is no you know it is a very smooth function or there is no discontinuity anywhere. However, the drawback of the van Genuchten model is that so the volumetric water content can never go to 0 at any given suction. If you look at the θ versus ψ, this decreases, and because of the nature of the equation asymptote, this will never approach to 0. So, this is a major issue using the van Genuchten model. All these models are very simple, one can generate different curves by varying the modelparameters like a, m, n. You can assume some values for these parameters and one can play with this on a spreadsheet. And you will understand, how each parameter influences, the nature of the SWCC curve. So, you would see that this would never become 0. It only approaches 0 at any given suction value, which is physically notcorrect, because, from a thermodynamic point of view, this has been observed that when the suction value is at 10 power 6 kPa that is 1000 mega Pascal, the water content should become 0. However, use in the van Genuchten model that does not happen.
So, there is another popular model proposed by Fredlund along with his co-investigator Xing proposed another SWCC model in 1994. In this model, the expression is θ equals to 1 minus log 1 plus ψ by ψ r divided by log 1 plus 10 power 6 by ψ r times θ s divided by log e that is exponential of 1 plus ψ by a whole power n whole power m. So, this is kind of similar to your van Genuchten equation. However, you have a log here and exponential form here exp of one here. And you have one coefficient or variable here before this one. Here ψ r is the suction corresponding to residual water content. And in Fredlund and Xing, they often mention that θ r determination is also difficult, therefore that also adds up as one of the fitting variables. So, however, in Fredlund and Xing model, you have another fitting variable that is ψ r suction corresponding to your water content.This value is often modeled along with other fitting parameters like a, n, and m or ψ r is substituted with 1500 kPa in some of the Fredlunds works. In some works ψ r is considered to be 3000 kPa also. So, in some papers, ψ r is also used as a variable, and which is also fitted along with other parameters. So, ψ r you can also assume, which equals to 1500 kPa. And such fixing of ψ r value is done often in some of the models like geo studio etcetera. So, in this particular model interesting part is that here you have 10 power 6 value.Therefore, all ψ value should be substituted on kPa. And this 10 power 6 kPa valueindicates that at ψ equals to 10 power 6 kPa. When it is plotted θ versus ψ 10 power 6 kPa, this curve is forced or the water content is forced to come to 0. So, when you substitute ψ equals to 10 power 6 kPa that is at suction equals to 10 power 6 kPa, so this value becomes 1, so 1 minus 1 is 0. So, θ is forced to come to 0, at theoretical value of at 10 power 6 kPa 1000 mega Pascal’s.So, therefore based on the theoretical observation or thermodynamic point of view, they observed that at 10 power 6 kPa the water content should go to 0. Therefore, based on that they modified the van Genuchten model to bring in that feature that the water content goes to 0 at 10 power 6 kPa. So, even though the model parameters here n, m, n, a, n, m are often understood to be flexible. The restriction we made on van Genuchten equation and van Genuchten model like n greater than 1 may be applicable in thisparticular case also. So, the advantages here also, you can generate a smooth SWCC similar to van Genuchten equation. And extra advantage may be that the water content is forced to 0 at very high suction values. Therefore, for clay soils where the suction range extends to such values because for clay soils the suction range extends to 10 power 6 kPa. However, there is a major disadvantage also often observed by our group in fact that because of this particular feature, often it is seen that. When θ verses ψ is plotted, the plots often go, and then because it has to go to 0 here. Often bimodal curves are obtained using the Fredlund Xing model. So, this is the soilwater characteristic curve is the which decreases, and after that which is forced to when you plot this. So, often this is seen that this value decreases because it has to approach. It should have gone, the curve should have approached directly here, but instead of thatbecause it has to go to 0 at 10 power 6 kPa. Often the curve behaves in a different manner. And often bimodal curves are obtained using Fredlund Xing model, often like this or sometimes it is like this such bimodal behavior is observed, which does not have any significance, however, because of the model restriction that θ should go to 0 at ψ equals to 10 power 6 kPa such discrepancies observed in the models. We have several other models like Campbell model, where θ equals to θ r plus θ s minus θ r times 1 plus ψ by ψ naught into exponential of minus ψ by ψ naught. Here ψ naught is the soil water potential at the inflection point on the curve, θ r is residual water content, and θ s is saturated water content. And Gardner’s expression, the gravimetric water content equals the saturated water content divided by 1 plus ‘a’ times ψ power n. Here a and n are fitting parameters. Similarly, the Brutsacrt model where ψ equals to a times Ws by W minus 1 whole power 1 by n. Here a and n are fitting parameters or model parameters and others are similar to here.And McKee and Bump model ψ equals a minus n log of W by W s. Here again a and n are fitting parameters, and W is water content at any given suction, and W s is saturated water content. So, not just these models, we have several tens of models that are available in the literature and often used. But, most often used models are only the three, which are discussed little elaborately that is Brooks Corey, van Genuchten, and Fredlund Xing. We have seen that the Brooks Corey model has a discontinuity at the bubbling pressure or the AEV or air entry value. Therefore, even though it is very simple, often in the modeling of partly saturated flows it is very difficult to use. When it comes to the van Genuchten model, which is a three-parameter model, which can generate a smooth soilwater characteristic curve. However, the water content only approaches to 0, but it does not become 0 even at a very high suction value. When it comes to the Fredlund model Fredlund Xing model, the feature it has a featureof feature for reducing the volumetric water content to 0 at suction value equals to 10 power 6 kPa. And this has a new model parameter such as a ψ r ψ suffix r, which is the suction value corresponding to residual water content, which needs to be either determined along with other fitting parameters such as a, m, and n or which can be fixed to certain value like 1500 kPa or 3000 kPa. Another important point here with the Fredlund-Xing model is its limitation that thiscannot be reversed or it cannot be written as ψ in terms of θ. Van Genuchten model can be reversed or inverse form can be written. But, Fredlund and Xing's model cannot be written as ψ of θ that is a major limitation. So, therefore with other hydraulic conductivity models, this integration is a little difficult that we will see now.
When comes to the hydraulic conductivity function models, as I explained that not many experimental techniques are available for the determination of hydraulic conductivity function in the laboratory as well as the field. Only the multi-step outflow technique is available that too it has a limitation to use more than 1500 kPa. So, this is mostly restricted to coarse-grained soils. So, hydraulic conductivity function is often determined from the soil-water characteristic curve data only. So, for that we need to understand, what different models we have. In hydraulic conductivity function, we have empirical models, macroscopic models, and statistical models. The empirical and microscopic models, these are simple functions, they may use K s in the expression and some curve fitting parameters. These are simple expressions like y equal to m x plus b such as linear expressions or something about the use for thehydraulic conductivity functions. And or the using the macroscopic behavior, they may see the similarity between HCF and SWCC based on that they utilize similarities between these parameters, and they used to predict. Generally, the empirical models and macroscopic models are fitting models for the existing hydraulic conductivity function data. When you have determined using the pressure plate apparatus the hydraulic conductivity data, then these models can be used to fit and determine the fitting parameters. On the other hand, the statistical models are advantages, because there is a theory behind it. So, the hydraulic conductivity of they assume. So, in the statistical models, if we assume that this is one cross-section wherethis is one point where you have several pores that are available. So, in between, you have soil grains. So, in such a scenario, so this is one, for example, you have several soil particles, which are placed in this particular manner, there is no overlap. So, this a one particular soilstructure. When I consider the cross-section at any given place, the cross-section is this you have several pores designated by say r 1, r 2, r 3, which are distributed. Here we consider, circular pores that exist. And then, we consider the probability of connecting one soil pore with their adjoining water-filled pore with some probability. And then, we can consider that if there is a probability that, this pore exists. In the next to the other pore in the next cross-section, what is the probability of that, because if you if there is a probability that there is a pore exist, then flow takes place through that. Using the Hagen Poiseuilles equation, so the which states that the hydraulic conductivity is a function of r square. And as we know, if you consider, circular pores r can be approximated as 2Ts by h times gamma w, because this is pressure. Pressure equals to 2Ts by r, so when we assume that the contact angle is 0. So, therefore hydraulicconductivity is a function of 1 over h square or the head suction head.So, this is how the function is related, when different probability functions are used you may get an expression like K equals to T s square by 2 mu rho w g times eψlon square by n square times 1 over h 1 square plus 3 by h 2 square plus 5 by h 3 square like that and 2n minus 1 by h n square. So, such expression would result in the hydraulic conductivity of unsaturated soils. Here, T s is surface tension, mu is viscosity rho w g, and eψlon is the dielectric, and n is porosity. So, here the different expression for hydraulic conductivity function based on statistical models will have K as a function of 1 over h 1 square plus 3 by h square 2 h 2 square like that you will have. So, essentially they indicate different pores, sizes you have in the soil mass. So, if you look at different available models, so the old one is the Richards model, which is as old as 1931. So, the model is a linear assumption. The hydraulic conductivity is inversely linearly dependent. So, a simple linear equation where a ψ plus b, which is used this is a simple empirical equation. And Gardner, he had 1958 had a given expression, which is k s, which uses saturated hydraulic conductivity. So, k s by 1 plus atimes ψ n, a and n are empirical parameters. So, whenever you have these empirical equations the fitting parameters need to be determined. These are fitting parameters; the fitting parameters need to be determined by considering the best fit between themeasured hydraulic conductivity data and the expressions.And, Brooks Corey's method, which is a macroscopic model, because which construct similarity between the soil-water characteristic curve and hydraulic conductivity function. And based on the similarity, where you have λ, which is related to n, so that is how it is used, and so this is a macroscopic model. Similarly, Campbell proposed another model, which is also a macroscopic model. So,these three are statistical models, we have many other statistical models. But, here I have given three such models, this is based on Hagen Poiseuille's expressions. Where Jackson model, where it considers k of θ i equals to saturated hydraulic conductivity times θ idivided by θ s. So, θ i is at any given point i value and θ i by θ s times if sigma j equals to 1 to m 2 j plus 1 minus 2 i times h j over minus 2 that means, 1 over h square, that is how we have used also using Hagen Poiseuilles approximation. This form is divided bysigma j equals 1 to m, 2 j minus 1 divided by h square. So, this form is derived based on Charles and George's expression. And here K is the hydraulic conductivity at any given θ fine. So, this is another model, which is a Burdine model. And here the expression is in termsof relative hydraulic conductivity, where relative hydraulic conductivity is hydraulic conductivity at any given water content divided by the saturated hydraulic conductivity k by k s, which equals to θ square times integral 0 to θ is normalized volumetric waterconductivity, 1 by h square x dx by 0 to 1 1 by h square x dx. Here x is the integration variable. The Mualem expression is also similar to the Burdine model. So, these two expressions Mualem and Burdine is Burdine are very often used in the geotechnical engineering aswell as in soil science literature also. Here for integrating or for estimating the hydraulic conductivity function from the soil-water characteristic over data. We require a continuous function of h of θ. Here if we have h of θ, so we can directly substitute it here and we get an expression for hydraulic conductivity function. So, here we do not require any other fitting parameters. What are the fitting parameters we established for the soil-water characteristic curve that can be directly used for the hydraulic conductivity function determination?. This is also seen for many coarse grind soils these models provide very satisfactory results. Here, if you see the van Genuchten model if you recollect your van Genuchten model, so that is θ equals to 1 by 1 plusalpha h power n and whole power m. So, here it is written θ equals to function of h, but this also can be written ‘h’ as a function of or another function of θ. So, the inverse isvery easy to do, because that is what is required for the estimation of hydraulic conductivity functions here. However, if I go back using Fredlund and Xing model, in this expression, it is not possible to represent ψ as a function of θ, because the inverse is not possible, so that is a major limitation of Fredlund and Xing model. So that is a reason, why they use a different expression for determination of hydraulic conductivity function, which is called Kunze model K r θ r to θ θ minus x by ψ square x dx divided by θ r 2 θ s θ s minus x byψ square x dx.