 OK, so in this video, we'll be talking about advection dispersion equation. This is the first time we actually were not talking about chemistry, but talking about the physical processes. So what do we have thinking about this is, let's say you have chemicals that are non-reactive, right? So the system you have, whether we use that as an example is a column. So think about, for example, you are doing an experiment. You pack up a column with sand, grains, and then you inject a chemical, let's say bromide, from the left into the right. So essentially, you would have this chemical species you inject for a short period of time. You can imagine this chemical will be moving along with the flow. Over time, it will eventually be moving out of the system with certain velocities, via the velocity here. And you're talking about here, for example, the length of this column is L. So we want to know, as you can imagine, the concentration of this chemical will change with time and with space, right? Depends on what time you're looking at the snapshot. This chemical might be here, might be here, and other places there's no chemical species, because we consider maybe, for example, we start with the clean water. So how do we solve, or how do we mathematically solve this kind of equation and get a solution so we know the concentration of this chemical species in different time and different locations? And this is what we are going to talk about, which is advection dispersion equation we called ADE without a reaction term. So ADE, when we think about, I'm not going to derive in detail of where is this equation coming from. In general, the equation are coming from these mass conservation principles. So if we look at these different terms of the ADE, the first term we called the mass accumulation term. And it should have the units of accumulation, like mass per time. It's how fast things are changing, right? And this C is the concentration of the chemical species in the water face. So C here is concentration of this, let's call that tracer, in water. So everything we are solving is for how much they have in the water, because that's what we really care about. And this should have the units of, for example, mass per volume. So the first term is mass accumulation. Second term here we call the vector transport. This is the process where the chemical kind of, almost you think about swimming, the chemical is a tracer essentially flow together with the water at the same speed as the water flow. So that's called a defective transport. And the last term is a dispersive transport term. It's somewhat similar to diffusion process. It's driven by, first of all, concentration gradient. But also in this type of past media that you have grains, they tend to have different grain size, different size of the pore space. Some spatial engineering that leads to different flow velocity, mixing processes that leads to the concentration difference in different places. So these are the three major terms in this equation. And for phi, so this phi is what we call porosity. It's how much space you have, how much pore space you have in a given volume. So the past media has both pore space and solid space. So this is how much pore space into a percentage per unit of total volume. This v is velocity, flow velocity. It's the linear velocity in the pore space. So it's different from the u, we usually call dorsi velocity. And the relationship between the two, typically we know that u will be equal to phi plus a times linear velocity. So that's the relationship between the two. And this dh is a very important term for this dispersive transport. This will be equal to, I talked about, it could be coming both from diffusion and the mechanical dispersion in past media. So we have this kind of two terms adding together. This is coming from diffusion in what? In past media, diffusion coefficient in past media. And then the second term is essentially taking into account the mechanical dispersion. And you can think about alpha is we called dispersivity, which is a parameter described how fast mechanical dispersion happens. And it's related to velocity. So the whole term is also related to velocity. So the fast, the flow, you actually have larger term of this. So we have this equation. All the terms are here. And usually when we solve this equation, there is analytical solution for this equation. If you give us the right initial bundle condition. When we numerically solving this, we will be discretizing this equation in time and space. And then you get a solution for that. But before we do that, typically, we will need two conditions. This is the first time we introduce the space dimension, x here. Before, when in the mineral diffusion precipitation, less than we actually introduce the time. So you notice here that this equation we cause this is a partial differential equation. Meaning it has two independent variables. One is the time, the other is space. So this is the first time we introduce the space dimension here. Now, in order to solve this equation, we need to know at t equal to 0, what are the concentrations? So this is initial condition. When t equal to 0, what are the concentrations? And usually this is given for a given system. And we also need to know at the two boundaries, x equal to 0, x equal to L, what are the conditions that is specified? Is it, for example, a no flow boundary or pure divactive or pure dispersive? These are different type of boundary conditions you can specify. But anyway, you will need to, in both initial condition boundary conditions, to solve this. And different type of condition will give you different solutions. Because it matters what is the concentration in the t equal to 0. If you already start with something high, you will see a different concentration versus, for example, at t equal to 0, you have clean water. OK, so in terms of solution, let's assume we have done all this work to solve the equation. What do we expect to see after we solve this equation? So I'm talking about a system. I'll be using an example system. So you will be injecting a kind of a short pulse of this chemical bromide into a system. So if you concept or you think about it, at t equal to 0 is somewhere here. Let's say, everywhere, it starts with clean water. So you would have, at some point here, let's say you have a little bit smeared at early time. And with time goes on, you will have one more smeared. But going in the direction of flow and moving along and maybe become more. But it should be the total mass will remain the same. But at the end, you will see kind of a wider and wider over time and over longer distance. So this is conceptual how you would think about this solution you expect to see. Now, when we think about from mathematical term, let's draw this. When after we solve it, let's say we look at the concentration as a function of distance. And what do you expect to see at different time? So first of all, let's say at initial time, you probably would see something like this. In here, you see a pulse of this chemical. So this is t at about 0, maybe a little bit past 0. But you think about over time, this pulse will be moved along. And then you should see different distributions. So this would be, let's say, at t equal to v times some small concentration, t, you will see at another place. But if this would become a little bit wider, smeared out, this is t1. And over time, as you are going further distance, this becomes more and more smooth. I'm not drawing accurately. This mass total mass will remain the same. But I don't think I'm drawing nicely in terms of the mass conservation. But in any case, you will see over time, total mass will remain the same. But then the center of this will move along. So the speed of this plume, how fast it will move over certain time will be determined by the speed. v will determine the speed of the center, or the rate of the center moving to downstream. But also, another case would be, we know there's this dh, which is a very important parameter as well. So if you think about two different situations, one is, let's say they both have the same v, but they would have different dh values. Let's say we have another case with much higher dh. You will probably see something like this, a wider distribution, still the same total mass, but it's much more spread. So this will be representing a large, I'm sorry, a small dh. The blue one would be representing a much larger dh. So the larger the dh, it will be more spread. So essentially, you can think about dh will be determining the width of the plume. So this is t1, t2. And over time, for example, there's a t3, we care about it should be more, even more spread out. This is t3. So if I wait long enough, this is going to spread a lot. So that's the type of solution you expect to see when you have pulse of injection of a chaser. Now just very briefly mention the characteristic time. There are several times we think is important, right? Why is the residence time? This is directly related to how fast the flow goes and how long it actually stays in this column. How long this chaser actually stays in this column. So this is what we call tau A is equals to the velocity times the length over mu. Or you can just say L over v. This is residence time. Another time is how long it takes for the dispersion process to uniform the whole concentration field. This is what we call tau D. It related to dispersion. So it's L squared divided by dh. You can almost think about this. If it's not open system, it would be the time how long it takes for diffusion to go through, to uniform, to homogenize the whole concentration field. And then at that time, we define this packing number is tau D over tau A. So it's a relative between these two times scale or the rate of flow versus the rate of dispersion. And this should have a unit of L, u, phi, dh. This is a dimensionless number. OK, so these times will be determining how fast you realize these are kind of grouping dimensionless numbers. The packing number will determine the relative importance of dispersion versus advection. Which in the homework, I asked you to do some exercise under different conditions with different packing numbers.