Groundwater Clean-Up
One way to clean up a pollutant, such as fuel, in groundwater is to use bacteria that will breakdown the substance for food. We can consider a simple model of this letting the pollutant flow in one dimension, and letting the bacteria eat the pollutant and grow.
A simple model of one dimensional flow with diffusion is to consider a series of cells. Each time step, a certain proportion of the pollutant in one cell flows into the next. This gives us something like
In a spreadsheet we can us something like
|
|
A |
B |
C |
D |
E |
F |
|
1 |
|
x |
0 |
=C1+delx |
=D1+delx |
=E1+delx |
|
2 |
t |
|
|
|
|
|
|
3 |
0 |
Pollutant |
1 |
0 |
0 |
0 |
|
4 |
=A3+delt |
|
=(1-r*delt)*C3 |
=(1-r*delt)*D3+r*delt*C3 |
=(1-r*delt)*E3+r*delt*D3 |
=(1-r*delt)*F3+r*delt*E3 |
|
5 |
=A4+delt |
|
=(1-r*delt)*C4 |
=(1-r*delt)*D4+r*delt*C4 |
=(1-r*delt)*E4+r*delt*D4 |
=(1-r*delt)*F4+r*delt*E4 |
|
6 |
=A5+delt |
|
=(1-r*delt)*C5 |
=(1-r*delt)*D5+r*delt*C5 |
=(1-r*delt)*E5+r*delt*D5 |
=(1-r*delt)*F5+r*delt*E5 |
Where delx, delt, and r are parameters we define.
Running this with real numbers we get
|
|
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
t |
|
|
|
|
|
|
|
|
|
0 |
Pollutant |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0.5 |
|
0.85 |
0.1275 |
0.019125 |
0.0028688 |
0.00043 |
6.45E-05 |
9.68E-06 |
|
1 |
|
0.7225 |
0.21675 |
0.04876875 |
0.0097538 |
0.001829 |
0.000329 |
5.76E-05 |
|
1.5 |
|
0.614125 |
0.27635625 |
0.08290688 |
0.0207267 |
0.004664 |
0.000979 |
0.000196 |
|
2 |
|
0.5220063 |
0.31320375 |
0.11745141 |
0.0352354 |
0.009249 |
0.00222 |
0.000499 |
|
2.5 |
|
0.4437053 |
0.332778984 |
0.14975054 |
0.0524127 |
0.015724 |
0.004245 |
0.001061 |
|
3 |
|
0.3771495 |
0.339434564 |
0.17820315 |
0.0712813 |
0.024057 |
0.007217 |
0.001985 |
|
3.5 |
|
0.3205771 |
0.336605943 |
0.20196357 |
0.0908836 |
0.034081 |
0.011247 |
0.003374 |
|
4 |
|
0.2724905 |
0.32698863 |
0.22071733 |
0.1103587 |
0.045523 |
0.016388 |
0.005326 |
|
4.5 |
|
0.2316169 |
0.312682877 |
0.23451216 |
0.1289817 |
0.058042 |
0.022636 |
0.007923 |
|
5 |
|
0.1968744 |
0.295311607 |
0.24363208 |
0.1461792 |
0.071262 |
0.02993 |
0.011224 |
|
5.5 |
|
0.1673432 |
0.276116352 |
0.24850472 |
0.1615281 |
0.084802 |
0.038161 |
0.015264 |
|
6 |
|
0.1422418 |
0.256035163 |
0.24963428 |
0.174744 |
0.098293 |
0.047181 |
0.020052 |
|
6.5 |
|
0.1209055 |
0.235765712 |
0.247554 |
0.1856655 |
0.111399 |
0.056814 |
0.025566 |
|
7 |
|
0.1027697 |
0.215816306 |
0.24279334 |
0.1942347 |
0.123825 |
0.066865 |
0.031761 |
|
7.5 |
|
0.0873542 |
0.196546993 |
0.23585639 |
0.2004779 |
0.135323 |
0.077134 |
0.038567 |
|
8 |
|
0.0742511 |
0.178202607 |
0.22720832 |
0.2044875 |
0.145697 |
0.087418 |
0.045895 |
|
8.5 |
|
0.0631134 |
0.160939229 |
0.21726796 |
0.2064046 |
0.154803 |
0.097526 |
0.053639 |
|
9 |
|
0.0536464 |
0.144845306 |
0.20640456 |
0.2064046 |
0.162544 |
0.107279 |
0.061685 |
|
9.5 |
|
0.0455994 |
0.129958428 |
0.19493764 |
0.2046845 |
0.168865 |
0.116517 |
0.06991 |
|
10 |
|
0.0387595 |
0.116278593 |
0.18313878 |
0.2014527 |
0.173753 |
0.125102 |
0.078189 |
|
10.5 |
|
0.0329456 |
0.103778644 |
0.17123476 |
0.19692 |
0.177228 |
0.132921 |
0.086399 |
|
11 |
|
0.0280038 |
0.092412412 |
0.15941141 |
0.1912937 |
0.179338 |
0.139884 |
0.094421 |
|
11.5 |
|
0.0238032 |
0.08212103 |
0.14781785 |
0.1847723 |
0.180153 |
0.145924 |
0.102147 |
|
12 |
|
0.0202327 |
0.072837783 |
0.13657084 |
0.1775421 |
0.179761 |
0.151 |
0.109475 |
Now we can add bacterial growth. Suppose the change in the bacterial population is proportional to the difference between the carrying capacity and the bacterial population
where the carrying capacity is a function of P that increases with P but levels off, such as
(this leaves a low level population when there is no pollution.) Subtracting the amount of food consumed we get the situation shown on sheet 1.