 Prvo, da na toga naprejste, je dobro koncentruzajne. In zelo, da naredajte vse v počke in objezno, da je to način. Pero, vzelo, da je vzelo, da vzelo in čeljče za medija, je nekaj ubičati v zelo. V inšli, da je vzelo v pročne, da je zelo češnja, je zelo za mnohje odbačne, hveno stajevit iz spesela mikroskopivali spesela vs. napotoviti palošenje. Hvala pravda je, da je tudi paredo v zelo, vič perc, kako izvajajo vse neset, napotoviti tudi, da je postavno v kontekstu, in časovati bakterijel in mihro organizmu. Načo ne mislim, da je izgleda, če je akorizmidium. Včasno, da je bilo v sajem, časovati sajem in generim, akorizmidium, časovati sajem, in je vsečo čaratrizati vzelo, kaj je vzelo, da se vsečo je vzelo. V tem sej kontekstu, smo počutili o simulaciju in sintetikovih modelov. Tudi, po vzpečenju predpravitva, vedno begrač narodne modeli zelo zbijaj, da se naprezentuje na ovoj načine na vršenji dolajšenju 3D problemu. Vzpečenje z vršenja vzpečenja vzpečenja z vršenjem, are mostly laminar flow and processes that are dominated by visuous force. So we neglect inertial force and the mean velocity of the flow is really low. But even if this mean velocity is really, really low, the velocity distribution exhibit a really broad range of value. So as you can see here, you have a different value among three, four, or also more order of magnitude, also at the microscopic scale. So in that case, this chaotic fashion that occurs in porous medium comes, of course, by the close boundary condition, by the boundary condition that puts some constraints to the flow field. So in this framework, we are focusing about the filtration process and under this assumption is only related to the particle that are transported. So there is an advection terms and the interaction, the particle that may interact with the solid matrix. Now, all the theory, all the classical theory related to the filtration that takes the name, like classical filtration theory, are based on the fact that the entire porous material can be mapped in a single sphere. And under this assumption, the behavior of the particle that are transported may be described by the classical advection dispersion, this case dispersion equation, plus the terms that dictates the kind of interaction of the particles with the solid matrix, that in this case is only represented by a single sphere. So the entry here, we saw the advection dispersion equation with the reaction terms. The reaction terms is interpreted with a sort of attachment rate and the definition of attachment rate in this case under the Eppel sphere model is only dictated by a single collector efficiency. What is it a single collector efficiency? A single collector efficiency can be computed by a deterministic study of all the trajectory that flow through this sphere and all the trajectory that flow under a certain area that may be go in contact with the sum of all the force involved to the grain collector surface. Now, what is clear is that from this model, from the classical filtration theory, is that in the case that we are looking on stationary deposition profile, so looking at really long time behavior of a continuous or also a instantaneous sharp injection of particles but in this case it is more appropriate with a continuous particle injection if you are looking on the deposition profile that occurs along the filter, along the porous medium. What we saw is that the decay of the deposit particle along the porous medium decay in an exponential way. So, which is the problem? The problem are well summarized in a review that Shapiro wrote on 2006 in which he explained that the fact is that the CFT, the classical filtration theory is based on a really strong assumption. The assumption is that they always are considering irreversial attachment rate, so a particle may be attached and then release is not considered, at least in this case, in the first way that the model was stored and also this deposition, this attachment rate is considered constant and they are going to neglect also all the therogeneity that comes from the forefield and also the therogeneity that comes from the kind of charge surface that both colloidal and the grains of the solemn mafrix are affected. Now, what is clear is that the CFT fails to describe filtration under unfavorable condition. Favorable condition means that the particles may be spent around the grain, but they may be also be released at certain time. So, the attachment is not reversible and also fails to describe the filtration in strongly therogenous porous medium in which way the CFT fails in the way that he predicts, as I say before, exponential deposition decay along the porous medium, but we saw that in many experimental data deposition is not an exponential decay, so it is a sort of hyperesponential power law decay and also fails to predict the breakthrough curve, so the curve at which substance are reaching a certain control plane. So, which is our idea to trick this problem related to the long time behavior of the breakthrough curve and also this anomalous deposition profile, we suggest two numerical methods that are related to the deterministic, sort of deterministic particle tracking simulation and more useful random work method. Now, I will illustrate which kind of particle simulation we performed and the steps are really easy. We choose a geometry. Here you can see that the grain are composed as a representing the solid mafrix, so we solve the stock's equation for a given geometry of the solid mafrix. We introduce around each grain a potential that means the kind of physics that is involved around the single grain. So, if it is pure actrative, if it is pure repulsive, if there is a competition between actative force and repulsive force and all this kind, all this scenario potential are related to the kind of chemistry that is involved in the fluid that is flowing and once we decide the kind of potential we just perform particle tracking. Here you can see some details about irreversible. In this case we apply an easy attractive field around each grain, and so here you can see some irreversible event. Now, performing a simulation of a pure actrative field, here we can see one example of a simulated deposition profile that decays along a porous medium. In this case we are simulating a continuous injection, a uniform injection on this side, so the flow is flowing in this side, and here there is a position to the position of the particle that has been attached. Now, if we perform a sort of brute analytic, brutal statistical analysis on this profile, what we get in this case is still an exponential decay profile. And this is clear because how we say before that the deviation from an exponential profile are due mostly to unfavorable attachment condition. So, this most, this really easy example of which the attachable rate is irreversible confirm in such a way the assumption of which the classical filtration theory works. So, the next step that we must introduce in our particle simulation is a more complex potential around a single grain, and here there is the comparison between our trajectory, our analysis trajectory and the trajectory that are deterministic solved from a irreversible attachment rate, and here we see that a more realistic potential that represent the fact that both grains and colloidal particle are screened by charge surface. So, here we play, this charge surface plays a role like a screen in which particle may be entraped not from a non-under an irreversible case, but from a quasi statico permanent or transient trapping time. So, it's quite easy to introduce this more complex potential, but the problem is that from a computational point of view we can think about whatever potential we can add on the single grain to best explain the particular chemical dynamics, but the problem still remains about the computational cost that Lagrange, point of view, that is simulating the particle tracking implies. And in addition, if you want to set a more useful model which give us the possibility to predict the position profile not only not over a single porous medium of some centimeters, but of a Darcy scale, so many meters or also kilometers is quite impossible to think about particle tracking, performant porous scale and going through a field scale. So, since we can be able to hypothesize the best potential, the best international potential along the grain, we need something else, we need a further model, a statistical model to let this upscaling available. But the particle tracking, the particle tracking is not so, is not, we can use at least the particle tracking because from a statistical point of view from the particle tracking we can get some information about the single attachment rate, the single collector efficiency that affect the single grain and remember that at the beginning the attachment rate was considered like a constant along the porous medium. In this case we can perform a statistical PDF on the single grain for the attachment rate and also the velocity property, the Lagrangian velocity property that comes out from particle tracking can be used from a random walk based model. Now, the random walk, every random walk model needs, of course to be Markovian under a certain scale and the more classic random walk is considered a time step that is fixed, this is always the same after which the dynamic lose the memory of what happens asap before and so can be considered Markovian. So, in practice what we can do, we perform a sample along the Lagrangian trajectory and there are two different ways to perform sample, to sample the Lagrangian trajectory and one is based on the most naive random walk which say that we have to sample on discrete time sequence and the other way to sample trajectory we can sample trajectory along space distance. Why we do that? We do a sample of a space, of a discrete space distance for one reason because it's clear that in strongly heterogeneous porous media it's not possible to find a typical time scale over which the process is Markovian and there is a reason because low velocity plays a role and so if a party holds his experience really low velocity he will sample over that time many, many times. So there is a bias of overestimation of the low velocity and the result of this bias that the process is not exactly chaotic but there is some correlation of a sampling of time among the velocity which suggests that the best way to think about a random walk process in a porous medium is not to think about discretization over time but discretization of a land scale and mostly it's quite natural to think about a land scale that is reasonable with a typical porer scale. So the next step, the forward model that was thought in the end of the starting from the end of the 19 years is what is called a continuous time random walk. A continuous time random walk is based on a fixed length jump but a continuous time probability for the single party to make the jump. So in this really simple example of a 1D we have a fixed length can be derived from the Lagrangian statistical properties of the flow field and here the weight in times is distributed in a different way along each jump. So those are the basic rules. Here there is the typical land scale and here there is the velocity that can be extracted from the Lagrangian statistical properties of the flow field. So once we introduce these really easy rules we saw that the continuous time random walk there is a movie and I don't know why it doesn't work but the mathematical tools of the continuous time are related to the jump probability of the single party holes. So jump probability means that the party hole can make a jump of a certain distance taking a certain time. This jump in probability defines in a natural way a resident probability with just a complementar probability and the resident probability let us to interpret the concentration of party holes in a certain volume so this Lagrangian put a few of party holes like the only probability of a party hole to be in a certain space and can be weight and must be weight for a certain time interval where the probability of all the party holes that may reach this sample volume are due from this kind of propagator, this kind of green function which of course are related to the property of party holes that are outside can make exactly that jump to reach the sample volume posed on the x-t position. Why the CTRW was used? It was clear that the CTRW generalizacion of a random work so think about a continuous probability distribution for the weight in time was demonstrated that was able to from a molecular point of view from a poor scale from a really metroscopic scale was able to explain both sub diffusion and super diffusion process managed by the exponent of the power law that was assumed that dictates the probability the jump probability in a heterogeneous medium and on a larger scale the CTRW puts the evidence of the difference between the classical breakthrough curve that is given by a standard gaussian profile and put an evidence this kind of long time long-tail behavior of the breakthrough curve on scale there are much bigger so this simulation was performed of a scale of darsian and field scale so this is why the CTRW seems to be the best choice to make this kind 5 minutes perfect so giving this model is quite easy to think about a certain reactive CTRW in which the jump property are related of course to the flow property so the transport is given by the flow properties since we are still working in a porous medium we still continue to assume a fixed length jump with a time probability distribution to make this jump and a random attachment rate that we can say like we can see that a random decay rate over which if the time was panting to make the next jump is much bigger is still bigger of the decay rate it causes that a particle effectively decay so has been attached to that position of course we have to make some assumption a priori since we don't have still any reasonable ideal of which kind of probability distribution an attachment rate must be satisfied given a certain porous medium in this case we saw that a still power law attachment rate and there is a reason why we chose a power law because many other authors suggest that a power law decay rate should be best fit this anomalous behavior for the deposition profile what we saw that with those simple rules for the CTRW still continue to give us an exponential decay profile this is a similar plot so it means that this is not enough and this is quite clear because this kind of dynamics that is just a competition between at that time and a jumping time still continue to assume a certain irreversible attachment rate because the first time that the particle doesn't survive the particle has been attached what we saw is that if we assume a certain correlation length among the variation of the attachment rate so what does it means that in a practical way since in this case since in this case we were drawing the same attachment rate at each step in this case we say that the attachment rate is the same over a certain length distribution there is an experimental way I've finished there are many experimental reasons why we can hypothesize this kind of correlation about the variation the attachment rate but what we saw is that the CTRW with the same rules of before plus the introduction of this correlation length is able to derive a power law the position decay along the porous medium and it's clear that this is not the only the only resolution that we can get because it was shown that the CTRW explained the long tail behavior of the breakthrough curve so maybe we can hypothesize that this model can couple both this anomalous effect related to the long time behavior of the breakthrough curve and this anomalous the position profile so the conclusion just as we are just summarizing what I saw before and since there is no time there were another movie by Ford of course it doesn't work since for the same reason