# Oil Discharge Trajectory Simulation at Selected Baltic Sea Waterway under Variability of Hydro-Meteorological Conditions

## Abstract

**:**

## 1. Introduction

## 2. Process of Changing Hydro-Meteorological Conditions

#### 2.1. Definition

_{-}Markov approach as a more flexible and adaptable alternative to the conventional Markov model. Unlike the Markov model, which assumes exponential distributions for conditional sojourn times in specific h-m states, the semi-Markov approach allows for non-exponential distributions. This means that the model can accommodate any distribution of sojourn times in the respective states, providing a more realistic and accurate representation of the process. As a result, the semi-Markov approach is considered more practical, offering a better depiction of the real-world conditions.

#### 2.2. Parameters Identification for Selected Baltic Sea Waterway Area

- the velocity and direction of the wind;
- the height of the sea water level and waves;
- the direction of the currents;
- any obstacles to visibility such as fog or icing.

_{i}, i = 1, 2, …, 6, were counted and presented in Table 2.

_{1}(0) = 405/680 ≈ 0.595, p

_{2}(0) = 0.349, p

_{3}(0) = p

_{4}(0) = 0, p

_{5}(0) = 0.04, p

_{6}(0) = 0.016;

_{12}= 1516/(1516 + 1 + 31) ≈ 0.98, p

_{15}= 0.02, p

_{21}= 0.7, p

_{25}= 0.3, p

_{32}= 0.92, p

_{35}= 0.08, p

_{45}= 1,

p

_{52}= 0.76, p

_{53}= 0.01, p

_{56}= 0.23, p

_{62}= 0.41, p

_{63}= 0.15, p

_{65}= 0.44.

_{kiki}

_{+ 1}denotes the process’ random conditional sojourn time at the state k

_{i}under the condition that its next transition will be to the state k

_{i}

_{+ 1}, i = 1, 2, …, n. For instance, the process is at the state k

_{1}= 1 (wh up to 2 m and small ws up to 17 m/s), but under the condition it will go to the state k

_{2}= 4 (the wave remains the same, but wind is changing). That change has happened once and the realisation time t

_{k}

_{1k2}= t

_{14}= 6 h. Next, we are at the state k

_{2}= 4 and the next transition was to the state k

_{3}= 5 (wave height increased up to 5 m plus there is a strong wind) and lasted for 3 h, which means t

_{k}

_{2k3}= t

_{45}= 3 h. An example of a process’s A(t) realization is presented in Figure 5.

_{kiki}

_{+1}(t) were verified based on the sufficiently large realization number t

_{kiki}

_{+1}, and the empirical distribution functions were determined for the numbers of realizations less than 30.

_{kiki+}

_{1}representing the conditional sojourn times, where {k

_{i}, k

_{i}

_{+1}} = {{1,2}, {1,5}, {2,1}, {2,5}, {5,2}, {5,6}, {6,2}, {6,5}} were significantly large, and after verification it was confirmed that they exhibit chimney distributions (θ

_{12}, θ

_{15}, θ

_{21}, θ

_{25}, θ

_{52}, θ

_{56}, θ

_{62}) and gamma distribution (θ

_{65}), while the distribution functions of θ

_{14}, θ

_{32}, θ

_{35}, θ

_{45}, θ

_{53}, θ

_{63}have the empirical distribution functions (Figure 6 and Figure 7). Formulae of the functions W

_{kiki}

_{+ 1}(t) are approximated to three decimal places, whereas in the graphs there are illustrated functions drawn without that approximation. The chimney distribution is introduced in a book [42], Chapter 2, page 58. This distribution got its name from chimneys because of the chimney look of the density function. The chimney distributions were used because they fit very well to the given data sets. However, a few distribution functions have the empirical distribution functions and none of the common distributions can fit well to the data (at least 30 distributions were checked with 1% to 10% critical interval). Moreover, some transitions to next states do not appear frequently and there were a few realisations. For instance, a transition between states 1

**→**4: wh up to 2 m and small ws up to 17 m/s that increases above that value without much change in the wave height happened once during the considered period of time. Thus, empirical distributions were used.

#### 2.3. Input Data for Simulation

- the experiment time T = 48 h;
- formula for generating the initial state coming from (1)

- formula for generating next states

## 3. Algorithm

_{i}, i = 1, 2, …, n, and has the normal distribution with the parameters that will be slightly modified [39,40]:

- expected values ${m}_{X}^{{k}_{i}}(t),$ ${m}_{Y}^{{k}_{i}}(t);$
- standard deviations ${\sigma}_{X}^{{k}_{i}}(t)$, ${\sigma}_{Y}^{{k}_{i}}(t)$;
- correlation coefficient ${\rho}_{XY}^{{k}_{i}}(t)$.

_{i}, i = 1, 2, …, n, is dependent on time t ∈ 〈0,T〉.

_{i}= 1, 2, …,~m, create a curve K

^{k}called an oil spill central point drift trend (CPDT)

_{i}

_{–1}+ 1)∆t, (s

_{i}

_{–1}+ 2)∆t, …, s

_{i}∆t, are given by

_{i}

_{–1}+ 1)∆t, s

_{i}∆t〉 at state k

_{i}, with s

_{i}that satisfy the inequality

_{kj kj}

_{+ 1}= ${W}_{{k}_{j}{k}_{j+1}}^{-1}$(h), j, = 1, 2, …,i, i = 1, 2, …, n, are the inverse functions of the conditional distribution functions W

_{kj kj}

_{+ 1}(t) and h is a number that is randomly generated over the interval 〈0, 1).

- c
^{2}= −2ln(1 − p); - expected values

- standard deviations

- correlation coefficients ${\rho}_{XY}^{{k}_{i}}(t)$,

_{i}= 1, 2, …, b

_{i}, b

_{i}= 1, 2, …, s

_{i}− s

_{i}

_{–1}, i = 1, 2, …, n.

## 4. Application

#### 4.1. Experiment

_{i}∈ {1, 2, …, 6}, i = 1, 2, …, n, create an oil spill CPDT composed of curves K

^{ki}, i = 1, 2, …, n [40,65,66]. According to (5), we have

- ${\sigma}_{X}^{{k}_{i}}(t)$ = ${\sigma}_{Y}^{{k}_{i}}(t)$ = ${\sigma}_{}^{{k}_{i}}(t)$ = 0.1 + 0.2∙t;
- ${\rho}_{XY}^{{k}_{i}}(t)$ = 0.8;
- ${r}^{{k}_{i}}(t)=$0.5 + 0.5∙t,

_{i}∈ {1, 2, …, 6}, i = 1, 2, …, n, that in the real practice should be statistically identified using the methods given in [40].

_{1}, where k

_{1}∈ {1, 2, …, 6}, is selected according to (3), resulting in k

_{1}(0.62) = 2.

_{1}= 2, we determine the next state k

_{2}, where k

_{2}∈ {1, 2, …, 6}, using Equation (4), i.e., k

_{2}(0.45) = 1.

_{1}= 2 and k

_{2}= 1, we generate the first realization t

_{kiki}

_{+ 1}= ${t}_{{k}_{1}{k}_{2}}^{(1)}$ of θ

_{k}

_{1k2}from an assigned probability distribution, as indicated in Figure 6 and Figure 7. In this example, ${t}_{{k}_{1}{k}_{2}}^{(1)}$ = t

_{21}= 6.78. Consequently, we have (s

_{1}− 1) < 6.78 ≤ s

_{1}.

_{1}= 7 and s

_{0}= 0, s

_{1}− s

_{0}= s

_{1}= 7. Subsequently, we compare the value of s

_{1}with time T, which is equal to 48. Observing that s

_{1}= 7, which is less than 48, we proceed to draw the sequence of domains accordingly:

_{1}= 1, 2, …, b

_{1}, b

_{1}= 1, 2, …, 7, composed of the elliptical sub-domains shown in Figure 10.

_{2}= 1 is proceeded by randomly generating another set of numbers, g ≅ 0.76 and h ≅ 0.30. We select state k

_{3}(0.76) = 2, and generate another realization, ${t}_{{k}_{2}{k}_{3}}^{(2)}$ = t

_{12}= 18.75, for the conditional sojourn time. Once we have these realizations, we calculate their sum:

_{2}− 1) < 25.53 ≤ s

_{2}, we deduce that s

_{2}= 26 and the difference s

_{2}− s

_{1}= 26 − 7 = 19. Next, we compare s

_{2}with time T and observe that s

_{2}= 26, which is less than 48. Consequently, we obtain the sequence:

_{2}= 1, 2, …, b

_{2}, b

_{2}= 1, 2, …, 19.

_{1}, s

_{2}) = (7, 26〉 is partly illustrated for selected t = 8 h, 9 h, 15 h, 26 h, as shown in Figure 11. We can notice that it consists of elliptical sub-domains with bigger radiuses than those shown in Figure 10.

_{4}(0.03) = 1, we then obtain a realization as ${t}_{{k}_{3}{k}_{4}}^{(3)}$ = 52.3. The entire sojourn time is:

_{3}− 1) < 77.83 ≤ s

_{3}.

_{3}is greater than the experimental time 48 h, we substitute and calculate the difference s

_{3}− s

_{2}= 48 − 26 = 22. Based on this, we accordingly proceed to draw the sequence

_{3}= 1, 2, …, b

_{3}, b

_{3}= 1, 2, …, 22.

_{3}, s

_{4}) = (26, 48〉 is partly illustrated for selected t = 27 h, 28 h, 38 h, 47 h in Figure 12. Figure 13 displays the final domain at t = 48 h. The sequence for varying h-m conditions after two days of the oil discharge is presented. The simulation time can be increased and for this purpose the algorithm should be continued, i.e., the process involves substituting i with j and subsequently drawing new random numbers g, h. This is conducted to select the states k

_{i}

_{+1}and generate different realizations ${t}_{{k}_{i}{k}_{i+1}}^{(i)}(h)$. This is iteratively repeated until the cumulative sum $\sum _{j=1}^{i}$t

_{kjkj}

_{+1}obtained from all the generated realizations ${t}_{{k}_{i}{k}_{i+1}}^{(i)}(h)$ reach a new increased experiment time T. Then, the necessary parameters can be calculated and by following the aforementioned iteration process, sequences of domains under varying h-m conditions can be obtained. In the experiment, the oil spill domain is represented by the cumulative sum of sub-domains of an ellipse shape.

#### 4.2. Discussion of Results and Further Developments

^{ki}, its equation, and the radiuses ${r}_{}^{{k}_{i}}(t)$ of the moving oil spill elliptical sub-domains. In the real practice, the unknown parameters should be statistically identified.

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Numbers of tanker spills (medium and large). Source: Own elaboration adapted with permission from Ref. [5].

**Figure 2.**Number of tanker spills and quantities of oil discharge (medium and large) in the recent years. Source: Own elaboration adapted with permission from Ref. [5].

**Figure 4.**Selected Baltic Sea waterway at the Bornholm Basin between two ports (blue line) with highlighted hydro-meteotological measurement points [55].

States | wh [m] | ws [m/s] |
---|---|---|

1 | 0–2 | 0–17 |

2 | 2–5 | 0–17 |

3 | 5–14 | 0–17 |

4 | 0–2 | 17–33 |

5 | 2–5 | 17–33 |

6 | 5–14 | 17–33 |

State | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|

n_{i} | 405 | 237 | 0 | 0 | 27 | 11 | 680 |

State | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1→ | – | 1516 | 0 | 1 | 31 | 0 |

2→ | 1001 | – | 0 | 0 | 435 | 0 |

3→ | 0 | 11 | – | 0 | 1 | 0 |

4→ | 0 | 0 | 0 | – | 1 | 0 |

5→ | 0 | 298 | 2 | 0 | – | 88 |

6→ | 0 | 45 | 16 | 0 | 48 | – |

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**MDPI and ACS Style**

Dąbrowska, E.
Oil Discharge Trajectory Simulation at Selected Baltic Sea Waterway under Variability of Hydro-Meteorological Conditions. *Water* **2023**, *15*, 1957.
https://doi.org/10.3390/w15101957

**AMA Style**

Dąbrowska E.
Oil Discharge Trajectory Simulation at Selected Baltic Sea Waterway under Variability of Hydro-Meteorological Conditions. *Water*. 2023; 15(10):1957.
https://doi.org/10.3390/w15101957

**Chicago/Turabian Style**

Dąbrowska, Ewa.
2023. "Oil Discharge Trajectory Simulation at Selected Baltic Sea Waterway under Variability of Hydro-Meteorological Conditions" *Water* 15, no. 10: 1957.
https://doi.org/10.3390/w15101957