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Mathematical Modeling


The final part of our project is determining where to place our biosensors. We can’t simply throw our biosensors in the river and expect them to detect toxins. Advective and diffusive processes dictate how toxins are transported in a river and if we don’t take these into account we could potentially place our biosensors in a spot where they would never detect any toxins! Why is this true? Imagine for a moment that there is a river and in this river there is a spot where chemicals are seeping into the river from the soil. As these chemicals seep into the river they are carried downstream by the river through a process called advection. In addition to this, as the chemicals are carried downstream they spread across the width of the river through a process called diffusion. There are several different types of diffusive processes that depend on what scale of diffusion you are looking at. The most pertinent diffusion process for us is turbulent diffusion. If we combine these two processes together the resulting spread of chemicals is shown in Figure 1. As you can see, depending on where we place our biosensor we might never detect any chemicals at all because the chemicals haven’t spread there. If we want to place our biosensors so that they will detect chemicals we need to place them at least a distance $L$ downstream of the chemical source. However, if we place our biosensors too far downstream of the chemical source then the concentration of chemicals will be too dilute for our device to measure. So its best to place our biosensor a distance $L$ away from the chemical source.


So for a given river width $W$, how do we find the distance $L$ to place our biosensor? One way to do that would be to go to the river in question, measure its height, width, length and volumetric flow rate and numerically solve the advection diffusion equation. However the Alberta Oil Sands are roughly 4,000 km away from Cornell so it was impossible for us to go there and measure the rivers. Instead we decided to use dimensional analysis to estimate what the value of $L$ would be for different rivers. To do this we first defined a time scale $\tau$ which is the time scale for advection. $\tau$ can be written as $$\tau = \frac{L}{u}$$ where $u$ is the average velocity of the river and $L$ is the length scale for advection. Similarly we defined a time scale for turbulent diffusion which can be written as $$\tau= \frac{l^2}{D}$$ where $l$ is the length scale for turbulent diffusion and $D$ is the turbulent diffusion coefficient. Since advection and diffusion are happening at the same time we can set both time scales equal to each other and solve for $L$ which yields $$L= \frac{ul^2}{D}$$ The average velocity $u$ can be rewritten as $$u = \frac{Q}{wh}$$ where $Q$ is the volumetric flow rate of the river, $w$ is the width of the river and $h$ is the depth of the river. Next, the turbulent diffusion coefficient can be rewritten as $$ D=kh \sqrt{ghs}$$ where $s$ is the slope of the river, $g$ is the acceleration due to gravity and $k$ is an experimentally determined constant. This constant differs for turbulent diffusion across a river (horizontal) and turbulent diffusion within a river (vertical). Since we want to place our biosensor at the point where the chemical is completely mixed in the river, we need to take into account both horizontal and vertical turbulent diffusion. Putting both of these definitions into our equation for $L$ gives us $$L= \frac{Ql^2}{kwh^2\sqrt{ghs}}$$ Finally, we need to define what $k$ and $l$ are. If we want to know how far downstream the chemicals will be mixed horizontally across the river ($L_Y$) then $l$ = $w$ and $k$ = 0.6 [1]. If we want to know how far downstream the chemicals will be mixed vertically in the river ($L_Z$) then $l$ = $h$ and $k$ = 0.067 [1]. The larger of these $L$ values will determine where we need to place our biosensor for a giver river. This gives us the following two equations $$L_Y= \frac{Qw}{0.6h^2\sqrt{ghs}}$$ $$L_Z= \frac{Q}{0.067w\sqrt{ghs}}$$

Calculation of Biosensor Placement

Now that we have equations for finding how far downstream we should place our biosensors it is time to plug in actually values. The values we need are the height, width, slope and the volumetric flow rate of the river. Since we could not measure these values ourselves we conducted research to see if environmental organizations monitoring the Alberta Oil Sands have tabulated these values. In particular the Alberta Government [2] and the Regional Monitoring Aquatics Program (RAMP) [3] have been conducting hydrologic tests on the Alberta Oil Sands for over a decade. Using data collected from these two independent sources we were able calculated $L_Y$ and $L_Z$ for several different rivers in the Alberta Oil Sands. The placement of these biosensors can be seen on the map below. In addition to these biosensors we’ve also marked on the map potential locations for our biosensors for rivers where we couldn’t get data on but are important nonetheless for quantifying the release of toxins into the Athabasca Oil Sands. These sites were chosen based on their proximity to tailing ponds and their similarity to the rivers for which we have data on. The determined locations are marked in green and the potential locations are marked in red.

Athabasca RiverTar RiverSteepbank RiverMuskeg RiverJackpine Creek
$L_Y$71.233 m 1.683*10^-4 m0.1639 m0.0091 m12.2495 m
$L_Z$198.621 m 0.2893 m22.1028 m9.5927 m1.582*10^-4 m