Nonlinear Advection–Diffusion–Reaction Phenomena Involved in the Evolution and Pumping of Oil in Open Sea: Modeling, Numerical Simulation and Validation Considering the Prestige and Oleg Naydenov Oil Spill Cases

Abstract

The main goal of this article is to improve upon a previous model used to simulate the evolution of oil spots in the open sea and the effect of a skimmer ship pumping oil out from the spots. The concentration of the pollutant is subject to the effects of wind and sea currents, diffusion, and the pumping action of a skimmer (i.e., cleaning) ship that follows a pre-assigned trajectory. This implies that the mathematical model is of the advection–diffusion–reaction type. A drawback of our previous model was that diffusion was propagating with infinite velocity; in this article, we use an improved model relying on a nonlinear diffusion term, implying that diffusion propagates with finite velocity. To reduce numerical diffusion when approximating the advection part of the model, we consider second order discretization schemes with nonlinear flux limiters. We consider also absorbing boundary conditions to insure accurate results near the boundary. To reduce CPU time we use an operator-splitting scheme for the time discretization. Finally, we also introduce the modeling of coastlines and dynamic sources of pollutant. The novel approach we advocate in this article is validated by comparing our numerical results with real life measurements from the Oleg Naydenov and the Prestige oil spills, which took place in Spain in 2015 and 2002, respectively.

Appendix: Review of the Piecewise Linear Schemes and of the Limiters

Suppose that we want to approximate the solution of the following 1-D equation

$$\begin{aligned} \begin{array}{rl} \dfrac{\partial c}{\partial t} + \dfrac{\partial (vc)}{\partial x}=0&\hbox {in } {\varTheta }\times ( 0,T ). \end{array} \end{aligned}$$

(18)

Here c is the concentration, v is the velocity and ${\varTheta }=[\underline{{\varTheta }},\overline{{\varTheta }}]$, where $\underline{{\varTheta }}$ and $\overline{{\varTheta }}$ belong both to $\mathbb {R}$, and are, respectively, the lower and upper boundaries of the interval ${\varTheta }$. Next, interval ${\varTheta }$ is decomposed into I finite volume cells (intervals here), that we denote by ${\varTheta }_i=[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]$. For simplicity of notation, we assume first that $v>0$ is constant and that the lengths of the intervals ${\varTheta }_i$ are all the same, and equal to $\Delta x$.

One way to obtain such an approximation is, for instance, to assume that in each finite volume ${\varTheta }_i$, with $i=1,\ldots ,I$, the concentration c is constant throughout the volume. This simplification allows to generate first order numerical schemes, such as the upwind scheme used in [2]. However, it was observed that this scheme produces a high level of artificial diffusion. To address this issue, it would be better to assume that the concentration within each volume ${\varTheta }_i$ is an affine function of the position (see [32]).

In this case, in ${\varTheta }_i$ at time $t_n$ the concentration can be linearly approximated by:

$$\begin{aligned} c(x,t_n)=c^n_i+\sigma _i^n (x-x_i) \hbox {, for } x \in {\varTheta }_i, \end{aligned}$$

(19)

where $x_i$ is the center of ${\varTheta }_i$, $c^n_i=c(x_i,t_n)$ and $\sigma _i^n$ is the slope of the linear approximation. We note that $\sigma _i^n$ can be defined in several ways. For instance

• $\sigma _i^n=\dfrac{c^{n}_{i+1}- c^{n}_{i-1}}{2\Delta x}$, in this case we obtain the Fromm method.

• $\sigma _i^n=\dfrac{c^{n}_{i}- c^{n}_{i-1}}{\Delta x}$, in this case we obtain the Beam–Warming method.

• $\sigma _i^n=\dfrac{c^{n}_{i+1}- c^{n}_{i}}{\Delta x}$, in this case we obtain the Lax–Wendroff method.

For these three cases, $c^n_i$ is equal to the average of $c(x,t_n)$ over ${\varTheta }_i$.

At the boundary $x_{i-\frac{1}{2}}$, the flux $f_{i-\frac{1}{2}}(t)$, with t in the time interval $[t_n,$ $t_{n+1}]$, is:

$$\begin{aligned} f_{i-\frac{1}{2}}(t)= & {} v c(x_{i-\frac{1}{2}},t)=v c(x_{i-\frac{1}{2}}-v(t-t_n),t_n)\\= & {} v c^n_{i-1}+ v \sigma _{i-1}^n \left( \frac{1}{2}\Delta x - v(t-t_n) \right) . \end{aligned}$$

At the boundary $x_{i+\frac{1}{2}}$, the flux $f_{i+\frac{1}{2}}(t)$, with t in the time interval $[t_n,$ $t_{n+1}]$, is:

$$\begin{aligned} f_{i+\frac{1}{2}}(t)= & {} v c(x_{i+\frac{1}{2}},t)=v c(x_{i+\frac{1}{2}}-v(t-t_n),t_n)\\= & {} v c^n_{i}+ v \sigma _{i}^n \left( \frac{1}{2}\Delta x - v(t-t_n) \right) . \end{aligned}$$

Thus, on the time interval $[t_n, t_{n+1}]$ the variation of concentration over the volume ${\varTheta }_i$ is given by

$$\begin{aligned} \dfrac{c^{n+1}_i-c^n_i}{\Delta t}=\dfrac{f^{n+\frac{1}{2}}_{i-\frac{1}{2}}-f^{n+\frac{1}{2}}_{i+\frac{1}{2}}}{\Delta x}, \end{aligned}$$

where $f^{n+\frac{1}{2}}_{i\pm \frac{1}{2}}=\dfrac{1}{\Delta t} {\int }_{t_n}^{t_{n+1}} f_{i\pm \frac{1}{2}}(t) \mathrm {d} t$ denotes the flux average during the time interval $[t_{n},t_{n+1}]$ which is similar to the flux at $\dfrac{t_{n+1}+t_{n}}{2}$.

Considering that

$$\begin{aligned} f^{n+\frac{1}{2}}_{i-\frac{1}{2}}-f^{n+\frac{1}{2}}_{i+\frac{1}{2}} = v (c^n_{i-1}-c^n_{i})+ \dfrac{1}{2}v( \sigma _{i-1}^n - \sigma _{i}^n) ( \Delta x - v\Delta t ), \end{aligned}$$

we obtain the following space-time discretization scheme:

$$\begin{aligned} c^{n+1}_i=c^n_i+\dfrac{\Delta t}{\Delta x} \left( v (c^n_{i-1}-c^n_{i})+ \dfrac{1}{2}v( \sigma _{i-1}^n - \sigma _{i}^n) ( \Delta x - v\Delta t ) \right) \end{aligned}$$

(20)

Now, we generalize scheme (20) to the case of non constant velocities $v: {\varTheta }\rightarrow \mathbb {R}$.

In this case

$$\begin{aligned} f^{n-\frac{1}{2}}_{i-\frac{1}{2}}= & {} \dfrac{1}{2}v_{i-\frac{1}{2}} \left[ (1+\beta _{i-\frac{1}{2}})c_{i-1}^n+(1-\beta _{i-\frac{1}{2}})c_{i}^n\right] \nonumber \\&+\, \dfrac{1}{4}|v_{i-\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i-\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \Delta x \left[ (1+\beta _{i-\frac{1}{2}})\sigma _{i-1}^n+(1-\beta _{i-\frac{1}{2}})\sigma _{i}^n\right] , \end{aligned}$$

(21)

and

$$\begin{aligned} f^{n+\frac{1}{2}}_{i+\frac{1}{2}}= & {} \dfrac{1}{2}v_{i+\frac{1}{2}} \left[ (1+\beta _{i+\frac{1}{2}})c_{i}^n+(1-\beta _{i+\frac{1}{2}})c_{i+1}^n\right] \nonumber \\&+\, \dfrac{1}{4}|v_{i+\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i+\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \Delta x \left[ (1+\beta _{i+\frac{1}{2}})\sigma _{i}^n+(1-\beta _{i+\frac{1}{2}})\sigma _{i+1}^n\right] , \end{aligned}$$

(22)

where $\beta _{i\pm \frac{1}{2}}=1$ if $v_{i\pm \frac{1}{2}}\ge 0$ or $=-1$ if $v_{i\pm \frac{1}{2}}<0$.

Thus, scheme (20) becomes:

$$\begin{aligned} c^{n+1}_i= & {} c^n_i+\dfrac{\Delta t}{\Delta x} \left( \dfrac{1}{2}v_{i+\frac{1}{2}} \left[ (1+\beta _{i+\frac{1}{2}})c_{i}^n+ (1-\beta _{i+\frac{1}{2}})c_{i+1}^n\right] \right. \nonumber \\&+\, \dfrac{1}{4}|v_{i-\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i-\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \Delta x \left[ (1+ \beta _{i-\frac{1}{2}}) \sigma _{i-1}^n+(1-\beta _{i-\frac{1}{2}})\sigma _{i}^n\right] \nonumber \\&-\, \dfrac{1}{2}v_{i+\frac{1}{2}} \left[ (1+\beta _{i+\frac{1}{2}}) c_{i}^n+(1-\beta _{i+\frac{1}{2}})c_{i+1}^n\right] \nonumber \\&\left. -\,\dfrac{1}{4}|v_{i+\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i+\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \Delta x \left[ (1+ \beta _{i+\frac{1}{2}}) \sigma _{i}^n+(1-\beta _{i+\frac{1}{2}})\sigma _{i+1}^n\right] \right) . \end{aligned}$$

(23)

The previous scheme (23) is known to be conservative but not necessarily monotonous [10, 18]. This non-monotonicity may produce numerical solutions with unrealistic oscillations. These oscillations are due to the high variation of the concentration slopes $\sigma _i ^n$ near jumps of the concentration. A way to measure these oscillations is to use the concept of Total Variation (TV) defined as

$$\begin{aligned} TV\left( \{c^n_i\}_{i=1}^I\right) =\sum _{i=1}^I |c^n_{i}-c^n_{i-1}|. \end{aligned}$$

We are interested in creating numerical schemes with the property of Total Variation Diminishing (TVD), that is $TV(\{c^n_i\}_{i=1}^I)\ge TV(\{c^{n+1}_i\}_{i=1}^I)$. That property ensures that the scheme will not develop oscillations. Thus, we now introduce a variation of the scheme (23) that guarantee TVD.

To do so, we introduce the concept of flux limiters. In (21), we replace $\Delta x \left[ (1+\beta _{i-\frac{1}{2}}) \sigma _{i-1}^n +(1-\beta _{i-\frac{1}{2}})\sigma _{i}^n\right] \hbox { by } \phi (r_{i-\frac{1}{2}}^n)(c_i^n-c_{i-1}^n),$ and we obtain

$$\begin{aligned} f^{n-\frac{1}{2}}_{i-\frac{1}{2}}= & {} \dfrac{1}{2}v_{i-\frac{1}{2}} \left[ (1+\beta _{i-\frac{1}{2}})c_{i-1}^n+(1-\beta _{i-\frac{1}{2}})c_{i}^n\right] \nonumber \\&+\, \dfrac{1}{4}|v_{i-\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i-\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \phi (r_{i-\frac{1}{2}}^n)(c_i^n-c_{i-1}^n), \end{aligned}$$

(24)

where $\phi (r)$ is called flux limiter and $r_{i-\frac{1}{2}}^n=\dfrac{c_{i-1}^n-c_{i-2}^n}{c_i^n-c_{i-1}^n}$ if $v_{i-\frac{1}{2}}\ge 0$ or $=\dfrac{c_{i+1}^n-c_{i}^n}{c_i^n-c_{i-1}^n}$ if $v_{i-\frac{1}{2}}<0$. In a similar way we can rewrite

$$\begin{aligned} f^{n+\frac{1}{2}}_{i+\frac{1}{2}}= & {} \dfrac{1}{2}v_{i+\frac{1}{2}} \left[ (1+\beta _{i+\frac{1}{2}})c_{i}^n+(1-\beta _{i+\frac{1}{2}})c_{i+1}^n\right] \nonumber \\&+\, \dfrac{1}{4}|v_{i+\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i+\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \phi (r_{i+\frac{1}{2}}^n)(c_{i+1}^n-c_{i}^n), \end{aligned}$$

(25)

where $r_{i-\frac{1}{2}}^n=\dfrac{c_{i}^n-c_{i-1}^n}{c_{i+1}^n-c_{i}^n}$ if $v_{i+\frac{1}{2}}\ge 0$ or $=\dfrac{c_{i+2}^n-c_{i+1}^n}{c_{i+1}^n-c_{i}^n}$ if $v_{i+\frac{1}{2}}<0$.

Then, scheme (23) can be rewritten as

$$\begin{aligned} c^{n+1}_i= & {} c^n_i+\dfrac{\Delta t}{\Delta x} \left( \dfrac{1}{2}v_{i+\frac{1}{2}} \left[ (1+\beta _{i+\frac{1}{2}}) c_{i}^n+(1-\beta _{i+\frac{1}{2}})c_{i+1}^n\right] \right. \nonumber \\&+ \,\dfrac{1}{4}|v_{i-\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i-\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \phi (r_{i-\frac{1}{2}}^n)(c_i^n-c_{i-1}^n) \nonumber \\&-\, \dfrac{1}{2}v_{i+\frac{1}{2}} \left[ (1+\beta _{i+\frac{1}{2}}) c_{i}^n+(1-\beta _{i+\frac{1}{2}})c_{i+1}^n\right] \nonumber \\&\left. -\, \dfrac{1}{4}|v_{i+\frac{1}{2}}|\left( 1-\left| \dfrac{v_{i+\frac{1}{2}}\Delta t}{\Delta x}\right| \right) \phi (r_{i+\frac{1}{2}}^n)(c_{i+1}^n-c_{i}^n) \right) . \end{aligned}$$

(26)

We note that if in scheme (26) we take:

• $\phi (r)=0$, we recover the first order upwind scheme (a scheme producing a high level of artificial diffusion).

• $\phi (r)=\dfrac{1}{2}(1+r)$, we recover the Fromm scheme.

• $\phi (r)=1$, we recover the Lax–Wendroff scheme.

• $\phi (r)=r$, we recover the Beam–Warming scheme.

The first scheme is first order accurate and TVD. The second, third and fourth schemes are second order accurate, but non-TVD.

Let us consider the following nonlinear flux limiters:

• $\phi (r)=$minmod(1, r), we obtain the minmod scheme [28].

• $\phi (r)=\max (0,\min (1,2r),\min (2,r))$, we obtain the superbee scheme [28].

• $\phi (r)=\max (0,\min ((1+r)/2,2,r))$, we recover the monotonized central scheme [33].

• $\phi (r)=(r+|r|)/(1+|r|)$, we recover the Van Leer scheme [32].

• $\phi (r)=(r^2+r)/(r^2+1)$, we recover the Van Albada 1 scheme [31].

The five above schemes are TVD.