A Strategy for Bioremediation of Marine Shorelines by Using Several Nutrient Release Points - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 26

Q i (

)

, such factors are chosen as the solution of a quadratic programming problem.
Also, we prove the existence and uniqueness of this optimization problem. Note that
a strategy is called optimal if it solves the problem and, at the same time, minimizes
the total mass introduced into the aquatic system to mitigate the impact of nutrients
on the marine environment and to reduce the remediation cost. Thus, by introducing
the least amount of nutrients, the optimal control not only cleans the zones, but also
protects the whole ecosystem.
The new strategy considers a few discharge points located so that each oil-polluted
zone contains just one discharge point. It generalizes the previous strategy where the
only source was used to distribute nutrient in all oil-contaminated areas. Analytical
and numerical results for the case of unique source were obtained by considering
variational formulations [ 31 ], quadratic programming problems [ 32 ] and linear pro-
gramming problems [ 33 ].
Taking into account all the above remarks, the variational problem of the optimal
two-stage remediation strategy is posed as follows:

t

T

N

1
2

Q i ( minimize m ( Q 1 ,..., Q N ) = t ) dt (2.1)

0

i = 1

T

1
˄ | ʩ i |

subject to: c i − ʱ i ≤ J i (ˆ) =

ʩ i ˆ(

r , t ) drdt ≤ c i + ʲ i , 1 ≤ i ≤ N

T − ˄

(2.2)

0 ≤ Q i ( t ) ,0 ≤ t ≤ T , 1 ≤ i ≤ N , (2.3)

where m is the functional that represents the total mass of nutrient released into the
aquatic system within a time interval [ 0 , T ] and ˆ = ˆ( r , t ) is the concentration of

this substance at point r in D at the time t 0. Such concentration will be determined

with a dispersion model described in Sect. 2.2 . Besides, the functional J i (ˆ)

>

is the

ʩ i within time interval [ − ˄, ]

mean concentration of nutrient in the i th zone T T

≤ ≤ )

(1 . Hereafter, we refer to this functional as the direct estimation of

nutrient concentration. Without loss of generality, all the zones

i N

ʩ i considered in this
chapter are nonintersecting. The constraints in Eq. ( 2.2 ) are imposed to maintain the
concentration J i (ˆ) in a vicinity of the critical concentration c i required for optimal

biodegradation (1 ≤ i ≤ N ). Thus, c i − ʱ i is the minimum concentration of nutrient

in the oil-polluted zone ʩ i acceptable for efficient stimulation of the biodegradation

process, while c i + ʲ i is the maximum allowable concentration of nutrient in the

oil-polluted zone ʩ i established for the protection of aquatic system. Note that

the introduction of small positive parameters ʲ i increases the number of

feasible solutions of problem ( 2.1 )-( 2.3 ), and therefore this problem is less restrictive
than that described by Parra-Guevara and Skiba [ 29 , 31 ]. Finally, we note that the
problem ( 2.1 )-( 2.3 ) can also be used to determine the optimal release parameters
in a fairly common case, when the repeated application of nutrients is required

ʱ i and