Advection-Dispersion Equation (ADE) for non-reactive tracers
So in is this video we'll be talking about advection-dispersion equation. This is first time actually we're 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 nonreactive. So the system you have we could use it as example is a column. So think about, for example, you are doing an experiment. You pack up a column, a column with like sand grains. And then you inject a chemical, let's say bromide, from the left into the right. So essentially you would have these chemical species you inject for a short period of time. You can imagine this is chemical will be moving along with the flow. So over time it will be eventually moving out of system with certain velocities. So v is the velocity here. And you are talking about here for example is the last of this column is L. So we want to know, as you can imagine, the constitution of this chemical would change with time and with space. Depends on what time you're looking at the snapshot. This chemical species might be here, might be here and other places there's no of this chemical species because we consider maybe, for example, we start with clean water.
So how do we mathematically solve this kind of equation and get the 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 the advection-dispersion equation. We call it ADE with the other reaction terms. So ADE, when we think about it, I'm not going to derive in detail of where is this equation coming from. In general these equations are coming from these mass conservation principles. So if we look at these terms, different terms of the ADE, so the first home that 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. And this C is the concentration of the chemical species in the water phase. 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 the mass accumulation. The second term here we call advective transport. So this is a process where there's a chemical almost you think about swimming. The chemical is a tracer. It actually flows together with the water at the same speed as the water flow. So that's called advective transport. And the last term is a dispersive transport. It's somewhat similar to diffusion process. It's driven by, first of all, concentration gradient, but also in this type of medias. You have grains. They tend to have different grain size, like different size of pore space and some spatial [INAUDIBLE] that leads to different flow velocity, mixing processes that lead to the concentration difference in different places. So these are three major terms in this equation. And for phi-- so this is phi is we call porosity, is how much space you have, how much pore space you have in a given volume. So the porous media has both pore space and solid space. So this is how much pore space in terms of percentage, per unit of total volume.
This v is velocity, flow velocity. It's a linear velocity in the pore space. So it's different from the u. We usually we call that's the velocity. And the relationship between the two, typically we knows that-- velocity-- we know that u will be equal to 5 porosity times the linear velocity. So there's a relationship between the two. And this Dh-- Dh is a very important term for this dispersive transport. This would be equal to-- I talk about it could be coming both from diffusion and the mechanical dispersion in porous media. So we have this kind of two terms adding together. This is coming from diffusion in porous media-- diffusion coefficient in porous media. And then the second term is essentially taking into account the mechanical dispersion. And you can think about alpha is called dispersivity, which is a parameter to describe how fast mechanical dispersion happens. And it's related to velocity. The whole term is also related to the velocity. So the flow, you actually have larger term of this.
So we have this equation. All the terms are here. And usually when this equation, there's an analytical solution for this equation if you give the right initial boundary condition. When we're numerically solving this, we will be discrete sizes of this equation in time and space, and then you get a solution for that. But before we do that, typical we need two conditions, right? This is the first time we introduced a space diminishing, x here. Before in the mineral dispersion precipitation, last time we actually induced the time. So you'll notice here that this equation, we call this is a partial differential equation, meaning it has two independent variables. One is the time. The other is space. So this is first time we introduce a space dimension here.
Now in order to solve this equation we need to know, like at t equal to 0, what are the concentrations? So this is initial condition. When t equal to 0, what are the concentrations? Now 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 advective, or pure dispersive? These are different types of boundary conditions you can specify. But you will need to, in both the initial condition boundary conditions, to solve this. And different types of condition will give you different solutions because it matters what is the concentration in the t equal to 0. If your already start with something high you will see a different concentration versus, for example, at t equal to 0 you have clean water.
So terms of a 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 kind of shot pulse of this chemical bromide into a system. So if you conceptually think about it, at t equal to 0 it's somewhere here. And let's say everywhere it started with clean water. So you will have, at some point here, let's say you have kind of a little bit smeared at other time. And as time goes on you will have more and more smeared, but going in the direction of flow and moving along and maybe become more. So total mass will remain the same, but at the end you will see kind of wider and wider over time and over a longer distance. So this is conceptually how you would think about this solution you expect to see. Now when we think about from mathematical term, let's draw this. After we solve it, let's see. We'll 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. And here you see a pulse of this chemical. So is t at about 0, maybe a little bit past 0. But you think about, OK, over time, this pulse will move along. And then you should see different distributions. So this would be, let's say, at t equal to v times some small consolation, tu with C at another place. But this will become a little bit wider, smeared out. This is t1. And over time as you're going further distance, this becomes like more and more smooth. And I'm not drawing accurate. So total mass will remain the same. And I don't think I'm drawing nicely in terms of mass conservation. But in any case you will see over time total mass remain the same. But then the center of this will move along. So it's a v. So if it's a speed of this plume, how fast it moves over a certain time will be determined by-- so speed, v determines the speed of the center, of the rate of the center moving to downstream. But also another case will be-- we notice this Dh, which is a very important parameter as well. So if you're think about two different situations, let's say they both have the same v, but they will have different Dh values. Let's say we have another case with much higher Dh. Your 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 your large-- I'm sorry, a small Dh. The blue one would be representing a much larger Dh.
So now I draw the Dh. It will be more spread. So essentially you can think about Dh determines the width of the plume. So this is t1, t2. And over time, for example, if there's a t3 we care about, it should be even more spread out. This is t3. So if we wait long enough, this is going to spread a lot. So that's the type of solution you would expect to see when you have pulse of injection of a tracer. Now I'm just very briefly mentioning the characteristic time. There are several times we think is important. Once is residence time. This is directly related to how fast the flow goes and how long it actually stays in this column, how long this tracer actually stays in this column. So this we call tau a is equal to porosity 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. So this is what we call tau d. It's related to dispersion. So it's L squared divided by Dh. You can almost think about this as like if it's not open system, it would be how long it takes for diffusion to uniform, to homogenize the whole concentration field. And then a lot of times we define this Peclet number is tau d over tau a. So it's a relative between these two time scales 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. So these times will determine like how fast-- you will realize this are kind of grouping dimensionless numbers. The Peclet number will determine the relative importance of dispersion versus advection, which in homework I asked you to do some exercises under different conditions with different Peclet numbers.
Print