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Hello everyone, we were looking at different soil-water characteristics of models and hydraulic conductivity functions, for the estimation of flow through Unsaturated Soils. So, we are preparing ourselves to take up the task of understanding the flow through unsaturated soils. Before we can get into the flow, we need to understand the SWCC models that are available for obtaining smooth soil-water characteristic curves and hydraulic conductivity function, well-established HCFs from the measured water attention data from the laboratory or the field and either by measuring the hydraulic
conductivity function data as well or prediction from the SWCC model. So, this whole process is done in this particular manner in the first stage we were obtaining the soil-water tension data as of which is the relationship between volumetric water content versus matrix suction. Often this data also we may not get directly, because we may need to use a different suction controlling or suction measurement techniques in the laboratory for the estimation of θ versus Ψ over a whole range ofsuctions. So, often we get in terms of maybe water content ‘w’. So, then using the relationship between ‘w’ and θ by using the density of soil, how it varies with either drawing or wetting knowing this data we can convert to θ. Once we convert to θ we can plot θ versus Ψ. So, we get a discrete data like this, the θ equals to θ s corresponds to very small suction or suction is close to 0. So, that is saturated volumetric water content which is porous soil, and as the suction increases the θ value decreases. So, this is the discrete data we obtain; however, this discrete data may not be directly useful and we require a smooth well-defined function between θ and Ψ, we utilize several existing retention models such as the van Genuchten model Van Genuchten model of 1980 or Fredlund and Xing 1984
model or any other model. You can use a Brooks Corey also BC model which is maybe 1964, but; however, with BC model we have measure issue that up to the air entry value, we have a θ equals θ s and beyond that, it decreases non-linearly. However, here the function is not smooth at air entry value. Therefore, we may not get a smooth function as we get with other models. So, when we use these models these are the retention models, we can obtain a smooth function like this using some optimization techniques that will be discussed now after this slide. So, using some optimization
techniques we can obtain a smooth function like this. So, then once we obtain a smooth function, using different statistical models such as a Mualem model are Burdin such statistical models one can use are even Fredlund Xing use Kunz model. So, suppose models could be used to obtain hydraulic conductivity function. In the last class in the previous class, we discussed how to obtain hydraulic conductivity functions by knowing the SWCC model parameters. The a, m, and n parameters in Van Genuchten are a, m n C r parameters in Fredlund Xing model. Once these parameters are
known using different statistical models like Mualem Burdin etcetera are Kunz. So, we have analytical solutions for a few cases and, for a few cases we have to solve numerically and we can obtain hydraulic conductivity function like this. So, once we have the SWCC model in the second stage and hydraulic conductivity function in the third stage using these two function functional forms, we can use to model the flow through unsaturated soils either it could be infiltration problem through soils, as it infiltrates the hydraulic conductivity starts increasing. So, if we have wettingdata, then we can directly utilize, and similarly, the suction value decreases because of
the suction grey that exists in the soil mass because of which there is a flow that is taking place. So, therefore, we can utilize this equation as well provided these are wetting data, then we can directly utilize to understand how the flow takes place through initially dry soil. Similarly, if you have a drying data drying SWCC and drying hydraulic conductivity function, then we can understand the evaporation behavior or drainage behavior through initially saturated soil. As I discussed in the previous classes that the one-dimensional flow through partly saturated soils can be understood using this kind of equation, which is similar to the consolidation equation. So, here you have the different variables θ, which is again a function of suction. So, ∂θ by ∂t equals ∂by ∂z of K hydraulic conductivity which is a function of suction therefore, is hydraulic conductivity function times ∂Ψ by ∂z plus 1. So, will see why we get plus 1 if you have a horizontal flow that is taking place then you
will not have this plus 1. So, if you have a vertical, then plus one is added that we will see in when we discuss about the flows. Say if you simplify this or to modify this equation to get the same dependent variables on either side of the equation, you can convert to suction throughout then you will have a
slope ∂θ by ∂Ψ times ∂Ψ by ∂t. So, this is the slope of the soil-water characteristic curvewhich is called the specific moisture capability, which equals to ∂by ∂z of hydraulic conductive function and this is the same as the earlier form. Now the K of Ψ m is hydraulic conductivity function and Ψ of θ is the soil-water characteristic curve. Here we require smooth functions to solve this equation Another important point here is the slope of the soil-water characteristic curve should be defined at every point; that means, the slope is the derivative of the soil-water characteristic curve, should be well defined. In the case of the Brooks Corey method, the ∂θ by ∂Ψ is not valid at air enter value because at that in fluxion point which is the very sharp inflection point at air enter value. So, therefore, this value is not valid. So,
therefore, that is the reason why we require very smooth functions like hydraulic conductive function and SWCCs to solve this particular equation to get θ volumetric water content as space and time. So, this is the data we are looking for. So, when we give this as an input we get this θ. How to get the soil-water characteristic curve from the measured retention data? We
have several optimization strategies like gradient-based optimization which are classical methods, to minimize the error between experimental data and theoretical data. So, here briefly discuss how this is generally done. From stage 1 that is the measured data measured water retention data to smooth soil-water characteristic curve how we obtain. So, generally, we build error functions are called objective functions, we build objectivefunction say which is a function of say x, x is the variable which we want to determine or
the model parameter we want to determine. So, how these functions, what is this function? The objective function can be built using the measured data. So, for example, this can be determined like this. So, the measured data difference between measured data and the theoretical data. So, say you have N number of points N number of measured data points. So, then volumetric water content theoretically determined for different suction values minus θ m that is measured by you can obtain square divided by the number of data points you have. So, this is the root mean square error they call. So, essentially when you have SWCC data like this, θ on yaxis and Ψ on the x axis. For different values of Ψ, you have, say N number of data points for corresponding N number of data points you can assume you assume one
particular VG model. So, which is maybe θ equals to θ r plus θ s minus θ r times 1 by 1 plus alpha h over n whole power m. Here we assume one set of alpha m and n here you can write in terms of ‘a’ also you can write a Ψ or Ψ by ‘a’, this can be written in the form of θ r plus θ s minus θ r times 1 over 1 plus Ψ by a Ψ by a power n whole power m. So, this is the general form that is used in geotechnical engineering. So, here when you use Ψ the a has units of kilo Pascal because Ψ has units of kilo Pascal here alpha has units of 1 over meter because ‘h’ is substituted in meters. So, therefore, either alpha m
and r simply a, m and n. So, you can assume one set of these model parameters, you can give a value to there is a this is related to air enter value if it is sandy soil you can give say 2 kilo Pascal e as the initial value. If it is a clay soil you can give a very high value say 1000 kilo Pascal or something and you can give some values to m and n may be n equals to 1.5 or something and m value you can give two or any value. If m and n are related are constrained, then you give the values accordingly. So, this is called the initial guess. So, this the initial guess that is given. So, when you give this
initial guess or you substitute these values, then you can obtain θ theoretical values. So, that may be somewhat like this. So, this may be theoretical and this may be experimental. This experimental or measured this is theoretical. So, then for any given suction value, so, there is an error, between measured and theoretical. So, this is what we are trying to minimize. You build an objective function that objective function is theerror function contains what is the error between this data point and theoretical data point this data point and theoretical data point you can use some of squares or root mean
square or mean square or any norm could be used to define an error function. Once error function is defined with respect to any given model parameter, say here the error function is the model parameter here is x could be a, m or n. If you have only one model parameter to determine, you may have an objective function like this. So, our objective is to minimize the objective function objective value or error function value and what is the corresponding x that is what we are trying to determine. So, this can be
solved using simple optimization. So, you can determine an initial guess maybe this is the initial guess xi and this is optimum; our objective is to get x optimum by assuming some x i; xi could be here or here or anywhere on this curve.m So, we use different gradient techniques, and using gradient techniques simple gradient technique is the objective function with respect to x should be 0 at optimum value. At the optimum value of x when x equals to the ‘x’ optimum this value should be 0 this objective function should be 0. See in case of the present thing which I discussing for example, if you have a dependent case where the objective function with respect to say a and another axis you have m and for example, if you have m and n restrain condition m and m and n are related. So, in that particular case, you have some functional form a three-dimensional form. So, where you have one particular model parameter where one particular set of model parameters a and m where the objective function is lowest are minimum. So, therefore,
do by dm and do by dn should approach to 0 at optimum value. So, that set of m and n we determine. So, we build an objective function here O in terms of a,m, and n if you have three parameters to determine are simply a and m only two parameters to determine then the objective function is minimized.
So, this is minimized there are several such optimization strategies using the gradient, where sometimes the utilize double derivative also and they solve. These are called classical optimization algorithms or strategies they utilize the gradient of the objective function with respect to the model parameters and they determine the optimum p
So, such strategy is used and software is built which is called RETC; to estimate the soilwater characteristic curve the smooth SWCC function from the measured data, where we can utilize the van Genuchten model or Brooks Corey model. And in Van Genuchten again you can utilize different restrain conditions a how m and n are related are m and n are independent and several such cases could be utilized and we can obtain a smooth function. And now I will demonstrate this is the free software free b, which is developed by Van Genuchten and Nieche aids etcetera and in 1991. There are several improved versions are available, one can simply download h there is a website called PC progress. If you give the keywords as pc progress at c in Google you will be guided to the webpage and you can download the RETC software which is a very simple software, I will demonstrate it now. So, this is the RETC software. So, initially, when you select a file, a new project here
you can give some name and you can give some descriptions at demo. Here it will ask you what are different problems type of problems, scale, units type of retention, or conductivity model soil hydraulic parameters. So, when you choose that type of problem, a new window opens up here it says welcome to RETC and type of fitting you want to do both retention data and conductivity or diffusion data. So, retention data is soil-water characteristic data and conductivity or diffusivity data.Generally, we do not have conductivity data I have some retention data for loom soil from Aria of Paris paper 1981. So, I will use that data to demonstrate how it fits using a gradient-based optimization algorithm called 11 Breg mark wet algorithm. So, I will select only retention data because I have only retention data and when I select next. I can choose different units, here the input for the suction should be done soil-water potential, that is suction head should be substituted, therefore, we select meters, I can substitute suction head in meters or centimeter or mm. So, generally, we have suction values very high suction values such as 500 kPa, 1000 kilo Pascal etcetera in for finegrained soil, which we deal in soil mechanics. So, therefore, it is better to select in
meters and it could be anything because if we are not dealing with time units now because we are not estimating hydraulic conductivity now.
So, here for the inverse problem if you have then the maximum number of iterations number of printed iterations you can choose I will leave everything default. So, here you have several models Van Genuchten variable m and n model. So, this I have discussed where you do not have any restriction between m and n, only the restriction is that n is select n is chosen to be more than 1, n less than or equal to one results into nonphysical data at a suction value close to 0. So, you have corresponding conductivity
models also you can choose, you can choose Mualem or Burdin. So, therefore, accordingly, m and n parameters are estimated by optimization, the hydraulic conductivity functions are also determined immediately.
So, you have Van Genuchten model, m equals to one minus 1 by n this is the particular solution. So, I have discussed this when we made k equals 0 you have a particular solution for that this is an analytical solution available. So, the corresponding analytical solution you can choose again Mualem model similarly in combination with Burdin they derived another solution k equals to 0, for which the hydraulic conductivity function has the particular solution, so, for which m equals to 1 minus2 by n. Similarly, you can choose Brooks Corey model or Brooks Corey along with Mualem and Burdin are lognormal distribution or dual-porosity models these models I did not discuss, but other
models I have discussed in the previous classes. So, you can choose any model we can choose Van Genuchten m equals to m minus 1 by n model, and correspondingly the Mualem can be selected for hydraulic conductivity
function estimation. So, then if you click next; it will ask you it will show you different parameters that are available that is θ r this is θ r the residual water content do we know this data are know if we do not have we can fit it θ s is the data is known, then you do not need to fit it. You do not need to fit it you can select this data. So, as of now leave it as it is, and alpha value I will fit, n value I fit and Ks if I do not have if I do not need to fit, then I can select
some value here I am leaving a default value like this. So, now you can choose from this menu different soils possible soils you are fitting for.
So, actually, I am fitting for loam, this information is also not required for the
optimization, but here this information is used. So that, the initial guess will be generated based on this data. You can recollect what I said for the gradient-based optimization techniques we require an initial guess. Based on initial guess using the gradient we move to the next step and reach the optimum value. So, how to chose initial guess is the big issue? See here the initial guess is chosen based on the type of soil. So, if you select loam I have the loam data. So, I select the loam. So, then initial guess will be taken
accordingly then if I select next. So, here I can input the pressure versus θ here pressure should be inputted in meters. So, I can input the data that is 13.94 meters pressure corresponding to θ value of 0.099 and pressure value of 4.255 and corresponding θ as pressure is decreasing the θ value should
increase. So, this is 0.12 pressure is 1.608. So, the θ data is 0.172 and this is 0.539, this value is 0.248 and this 0.0953 and this data is 0.364 and this 0.0 0.0539 and this value is 0.39 and this is 0.0181 and this value is 0.42 and 0.06, this value is 0.434. So, this is the data we have. So, here you can give weightage based on whether the measured data is accurate or not, there may be slight chances of when you do the test several times. So, this the
measured data therefore, there may be some small error that could be there. So, you can give weightage whether this data is reliable or this particular data set is data point is reliable or not. So, here I am not giving any weight or anything I am just leaving the default values. So, then when I go for the next step, do you want to save the input data before executing a RETC I select. Then it asks you to do you want to run the application yes. So, then it shows you this one and you need to give enter. So, once you enter. So, you have the output file. This is the fitting that has been done. So, here the water content is drawn on the y-axis and pressure head. So, that is in meters is plotted on the x-axis. So, these are the data points. So, these red marks are the data points that we have inputted. And this line is fitted line and this is hydraulic conductivity versus pressure head. So, which we cannot fit here in log scale if you see log conductivity versus pressure head, this is how the hydraulic conductivity varies with the suction head; as a pressure head increases the hydraulic conductivity decreases. So, this is the output file and if you see the graph, this is the water content versus pressure head which is well fitted. So, these red marks are the data points inputted and this is this black line is the fitted curve. So, when you plot the conductivity in meter perday and log scale and pressure head in meters if you plot as a pressure head increases the
conductivity decreases. So, this is the plot you have. And we can see the RETC output file, which contains the information of what we used. We used Mualem based restriction that is m equals 1 minus 1 by N used. And this is the initial values of the coefficients θ R, we gave 0.078 and θ S 0.443. So,
this initial value is not fitted, but this is selected as an initial guess. But 0.43 this valuewe have inputted and alpha n m are taken as initial guesses and case we have given as 0.2496. So, then after the analysis, the r square we got is 0.987 very good r square value. So, non-linear least square analysis is done, and finally, this is the output. The θ R is estimated to be 0.0535 and alpha is 7 and n is 1.48 as n and m restricted case. So, once we get the ‘n’ we can determine the ‘m’ as well. So, this is the data we get. So, once we get the ‘m’ and m are n and alpha values and θ R. So, these three parameters unknown parameters are known and we have the other known parameters like θ s and other data we can use and plot it in excel on our own and that is how the SWCC smooth function is obtained by RETC software. So, this is how RETC free software could be used for the estimation of the smooth soilwater characteristic curve. So, we can also obtain the hydraulic conductivity function. However, if you are interested in using Fredlund and Xing model to the RETC software, does not allow you to use Fredlung Xing. So, then you need to use any other software like GeoStudio, but these are commercial software, where you can generate or you can write simple code in MATLAB and you can optimize.
So, the gradient-based algorithms have some issues such as the initial guess needs to be thoroughly examined, because generally, the objective functions may not be as smooth as I have shown So, the objective function could be like this, but because objective function versus say x, but because we are using some experimental data measured data there may be some error. So, this may not be as smooth as this you may have some roughness and you may have data like this. So, in that particular case, so, this may be your global minimum, but your function may get stuck here itself at local minima itself. So, if you chose an initial guess somewhere here, this may be your initial guess and it will traverse and it will get stuck here itself it may not reach here or it may be0 here itself. So, the objective function gets zero at the local minimum value. So, therefore, the estimated parameters may be erroneous. So, therefore, there are some global optimization algorithms. So, those are based on nature’s inspiration. So, such algorithms are genetic algorithm, ant colony optimization, particle swarm optimization in that there are many variations of course, in genetic algorithm also there are many variations and ant colony also there may be few variations and particles swarm optimization there are many variations like quantum behaved PSO, perturbed PSO which is our work from our research lab and bee colony algorithms etcetera.So, these algorithms very limited algorithms have listed here, but there may be an enormous number of algorithms based on the observations in nature. So, here for example, if I describe ant colony the way the ants move from their colony to the food sources by optimizing the path between the colony and the food source, that is one of the inspirations for building an optimization algorithm, there is an optimization involved in doing that in the social behavior of ants, so, based on that the algorithms were built.
Similarly, the particles formed to optimization are based on the behavior that is exhibited by the flocky birds and fish colonies etcetera. So, birds try to fly in a very formed shaped and there will be a leader and followers and they usually form this v shape and they fly to minimize that drag force. So, there are several rules to maintain such v shape and then while flying. So, such
optimization behavior is brought into the algorithm and which is used for solving several engineering problems. So, for example, the particles swarm optimization, which has a simple rule, for example, you have two parameters x 1 and x 2 you can choose initially some number of solutions on the shear space this is the shear space. The x 1 in our case could be a and x 2 ij our
case could be m or n. So, two-parameter optimization problem we have this search space we know the limit limits of the ‘search space’ as well because a can vary only betweensome particular range say 0 1 kPa to minimum 1 kPa to very large value like 5000 kPa 10000 kPa because this is related to the air-entry value. And m also m and n can be restricted in a given range. Now, a possible number of solutions are chosen on the shear space and each solution indicates a set of a 1 and m 1 and you chose another one this has a 2 and m 2. These are the two possible solutions for our problem and similarly, you have chosen n number of such solutions on a search
space i equals to 1 and this is the i equals to 2 and similarly you have n number of solutions. Now, i equals 1 corresponding to this you can obtain the objective function O 1, and corresponding to i 2 you can obtain the objective function O 2. So, now, one you have once you have the objective function O 2 and O 1 and O 2 you can compare these two. And if the objective function is lower, this value is lower than O 2 which means, the
global minima could be close to this particular value close to. So, what you could do is you can make this particle to move in this direction in the
direction of 1. So, one possible way of doing that is, for example, you have one particular particle here which is at distance say 0 and 2 and another particle are we call particles these number of solutions are called particles it could be birds also. So, that means, here I have another particle which is located at say 4 and say 5. How to move this particle to this value? So, directly to move this particle to here, what you could do is the difference could be added to this value and when it will jump into there. So, for example, the difference is of 4 and the difference is 3 if you add this point to this and this point to this it can directly jump here but we do not want to do that
directly it should not jump. Because this may not be the global solution, it should only move in this direction. So, for that what you could do? You can multiply with some factor. So, see θ or alpha or something. So, this is less than 1. So, then it can jump to a new solution in that particular direction. So, similarly, you compute the objective functions of all different particles, and whichever has the lowest objective function, all different particles will move towards that particle, for example, this particle has the lowest out of all the different particles, then all different particles will start moving towards him.
So, one iteration is complete, in the next iteration they have new positions. So, again you can compare all the objective functions and you can obtain what is the global best andalso because they have their own experience to move from here to here, they can compare within their history who is best this position is best or this position is best based on the local best and global best the particle will start moving towards the optimal solution and finally, at the end of the iteration once the search is complete all the particles will converge to one particular point that is a local minimum. So, I will
demonstrate this. So, this is one particular problem for the diffusion problem, where I have two parameters to optimize that is diffusivity diffusion coefficient and retardation factor, this of containment flow through saturated soils for that particular problem. So, initially, I have chosen a number of particles on search space, say these are the number of particles and they started searching and moving towards the global. So, this is initially the best solution. So, everybody started following this fellow and then they started coming close to and this also started searching and found that there is a better solution in and around. So, it also started perturbing, and finally, all the particles at the end of the search process they converge to a global solution.
So, that is how we search the global solutions using swarm intelligence or natureinspired algorithms. So, there are many such algorithms. So, one can use it to determine. So, once you get the smooth soil-water characteristic curve using optimization techniques, we can determine the hydraulic conductivity function that is stage 2 is SWCC, obtaining smooth SWCC stage 3 is obtaining K function. So, once SWCC isobtained the k function can be determined using either analytical solutions if there available or numerical solutions. If Van Genuchten model with some restrictions are chosen m equals one minus one by n or something then you are using the Mualem model then corresponding hydraulic conductivity function is known.
The solution is known as we discussed earlier and you can obtain, single Van Genuchten model using the Burdin’s equation is known then we can utilize the particular solution which was derived and we can get the hydraulic conductivity function. Similarly, if you are using m n variable case, so, then corresponding K r in terms of incomplete beta function could be used to obtain the hydraulic conductivity function. Similarly, several numbers of hydraulic conductivity functions can be determined based on what type of SWCC modeling is utilized. This is how we obtain hydraulic conductivity function. So, once we have SWCC and hydraulic conductivity function, once we have these two functions well-defined functions, we can utilize them for the flow problems. So, once we enter into the flow we can understand how these functions can be utilized and solved. Thank you.