Influence of Spill Pressure and Saturation on the Migration and Distribution of Diesel Oil Contaminant in Unconfined Aquifers Using Three-Dimensional Numerical Simulations

Abstract

Spilled hydrocarbons released from oil pipeline accidents can result in long-term environmental contamination and significant damage to habitats. In this regard, evaluating actions in response to vulnerability scenarios is fundamental to emergency management and groundwater integrity. To this end, understanding the trajectories and their influence on the various parameters and characteristics of the contaminant’s fate through accurate numerical simulations can aid in developing a rapid remediation strategy. This paper develops a numerical model using the CactusHydro code, which is based on a high-resolution shock-capturing (HRSC) conservative method that accurately follows sharp discontinuities and temporal dynamics for a three-phase fluid flow. We analyze nine different emergency scenarios that represent the breaking of a diesel oil onshore pipeline in a porous medium. These scenarios encompass conditions such as dry season rupture, rainfall-induced saturation, and varying pipeline failure pressures. The influence of the spilled oil pressure and water saturation in the unsaturated zone is analyzed by following the saturation contour profiles of the three-phase fluid flow. We follow with the high-accuracy formation of shock fronts of the advective part of the migration. Additionally, the mass distribution of the expelled contaminant along the porous medium during the emergency is analyzed and quantified for the various scenarios. The results obtained indicate that the aquifer contamination strongly depends on the pressure outflow in the vertical flow. For a fixed pressure value, as water saturation increases, the mass of contaminant decreases, while the contamination speed increases, allowing the contaminant to reach extended areas. This study suggests that, even for LNAPLs, the distribution of leaked oil depends strongly on the spill pressure. If the pressure reaches 20 atm at the time of pipeline failure, then contamination may extend as deep as two meters below the water table. Additionally, different seasonal conditions can influence the spread of contaminants. This insight could directly inform guidelines and remediation measures for spill accidents. The CactusHydro code is a valuable tool for such applications.

1. Introduction

In recent years, there has been an increasing interest in environmental safety and detecting pollution in groundwater and drinking water caused by hydrocarbon and crude oil leaks into porous media [1,2,3,4,5,6,7]. Most of these contaminants are light nonaqueous phase liquids (LNAPLs), which are less dense than water and are either immiscible or slightly soluble in water. They tend to distribute in both the unsaturated and the groundwater zone [3,8,9,10]. The quantitative and accurate determination of contaminant migration in space and time is a critical issue for implementing rapid remediation activities at contaminated sites [11,12,13,14].

The contamination impact of spilled hydrocarbons from an onshore pipeline can cause massive environmental damage. Pipeline corrosion can potentially lead to dramatic failures but also fatigue performance of corroded pipes with void defects [15,16]. Typically, these hydrocarbons are subjected to high pressures to facilitate their flow through the pipeline. Consequently, if the pipeline breaks, then the oil will come out at high pressure before interrupting the system, and the contamination migration will likely increase proportionally to the pressure inside the pipeline [17]. There are several techniques and methods for leak detection. Indeed, leaks and spills can occur in existing pipelines, and early detection of a spilled hydrocarbon is crucial to prevent significant damage to the ecosystem and life [10,18,19].

The migration and distribution of spilled oil contaminants can be numerically investigated using the equations of immiscible three-phase fluid flow in a porous medium. Their numerical resolution is challenging due to the nonlinear dependency on soil parameters, as the capillary pressure and permeability of each fluid phase depend on the saturation itself [20]. Additionally, the gravity term and pressure gradients are responsible for the creation of shock fronts and rarefactions. If they are not adequately treated, then they can cause significant numerical errors in the simulations and lead to inaccuracies.

Numerical models have been investigated, for example, to provide a quantitative assessment of LNAPL migration using T2VOC and comparison with experimental results [21], showing that LNAPL migration depends on the fluid properties, permeabilities, subsurface conditions and retardation [9,13], and water table fluctuations [22,23]. Accidental spills and leakages from underground storage tanks using FEFLOW [24], and migration of LNAPL in fractures filled with porous media [25], as well as transport and biodegradation combining the Hydrocarbon Spill Screening Model (HSSM) in the vadose zone and modified MT3DMS [26,27,28]. Insights for groundwater quality and pollution risk assessment, remediation of contaminated aquifers, enhanced delivery of remedial reagents in low-permeability aquifers from unsaturated soil to saturated aquifer have been investigated in Refs. [29,30,31,32,33,34].

This paper examines the impact of hydrocarbon pipeline pressure and water saturation in the vadose zone during a diesel oil spill. We follow the contours of saturation profiles of the three-phase fluid flow during migration in the variably saturated zone with great accuracy using the CactusHydro code introduced in Refs. [35,36] and validated through a sandbox laboratory experiment [37] and various applications that include free PCE extraction in potential emergency scenarios [38]. Also, an application to multilayered aquifers [39]. CactusHydro code employs a high-resolution shock-capturing (HRSC) flux-conservative method [40,41,42] that treats the advective part of the fluid flow equation better than other numerical methods (e.g., T2VOC, FEFLOW) when there is a discontinuity, since it is a conservative flux method. CactusHydro follows sharp discontinuities and temporal dynamics with high accuracy. On the other hand, a model limitation is that at this stage, we consider these fluids immiscible, and the process of volatilization and/or dissolution is not considered.

We present nine new scenario results with respect to a previous work [43]. We demonstrate that CactusHydro is capable of following the advective part of the multiphase flow with high resolution, including the observed shock fronts, which is the most critical component in vertical migration. Additionally, it captures the parabolic part that depends on the capillary pressures of the three-phase fluid flow components. We show contour saturation in the y–x plane for all scenarios Additionally, the mass distribution of the contaminant along the porous medium expelled during the emergency is analyzed and quantified in the different scenarios.

This study contributes to the understanding of the fate of migration from an oil spill under various hydrogeological conditions, some of which have not been previously studied. These results can serve as a valuable tool for implementing pollution control at contaminated sites and simultaneously preserving sustainable methods in hydrocarbon remediation.

2. Materials and Methods

2.1. Hydrogeological and Geological Data

The area of interest is situated in the central–eastern part of Italy, specifically in the Abruzzo Region, located between the Apennine chain and the foothills to the southwest, and the Adriatic Sea to the northeast, as discussed in Ref. [44]. The stratigraphic succession on the eastern side of the Apennines is discussed in [45,46,47,48]. This succession is dominated by a regressive marine depositional sequence that evolves from shallow marine to foreshore units, ultimately forming a coastal lagoon with alluvial channeling deposits [42]. Figure 1 shows the study area [44]. It illustrates the various hydrogeological complexes, including fluvio-lacustrine, sandy-conglomeratic, and clay-prevalent [49].

2.2. Mathematical Formulation

The numerical model is based on a conceptual model that describes an extreme emergency caused by a complete break of the onshore pipeline within an unconfined aquifer system. The pipeline is situated below the ground surface in the unsaturated zone at two meters above the water table. Just before the break, the immiscible fluid moves inside the pipeline with a fixed pressure value. Soon after the break, the hydrocarbon is released at the same pressure for one hour. After that, the system is shut down, and the leakage stops. The spill migrates through the unsaturated zone before arriving at the groundwater table. Being an LNAPL remains in the fringe zone due to a capillary pressure between the three phases: oil, water, and air. Due to a hydraulic gradient, the immiscible contaminant moves together with the fluid flow.

The mathematical formulation that describes this multiphase fluid flow composed of water $\left(w\right),$ contaminant $\left(n\right),$ and air $\left(a\right)$ in a porous medium is calculated using the conservation equation for mass and the Darcy law. We briefly review these equations to fix the notation that will be helpful in the following sections [35,36],

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_n\varphi S_n\right)={\displaystyle \frac{\partial }{\partial x^i}}\left[{\rho }_n{\displaystyle \frac{k_{rn}}{{\mu }_n}}{k}^{ij}\left({\displaystyle \frac{\partial p_a}{\partial x^j}}+{\rho }_{n}g{\displaystyle \frac{\partial z}{\partial x^j}}\right)\right]-{\displaystyle \frac{\partial }{\partial x^i}}\left[{\rho }_n{\displaystyle \frac{k_{rn}}{{\mu }_n}}{k}^{ij}\left({\displaystyle \frac{\partial p_{can}}{\partial x^j}}\right)\right]$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_w\varphi S_w\right)={\displaystyle \frac{\partial }{\partial x^i}}\left[{\rho }_w{\displaystyle \frac{k_{rw}}{{\mu }_w}}{k}^{ij}\left({\displaystyle \frac{\partial p_a}{\partial x^j}}+{\rho }_{w}g{\displaystyle \frac{\partial z}{\partial x^j}}\right)\right]-{\displaystyle \frac{\partial }{\partial x^i}}\left[{\rho }_w{\displaystyle \frac{k_{rw}}{{\mu }_w}}{k}^{ij}\left({\displaystyle \frac{\partial p_{caw}}{\partial x^j}}\right)\right]$$

(2)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_a\varphi S_a\right)={\displaystyle \frac{\partial }{\partial x^i}}\left[{\rho }_a{\displaystyle \frac{k_{ra}}{{\mu }_a}}{k}^{ij}\left({\displaystyle \frac{\partial p_a}{\partial x^j}}+{\rho }_{a}g{\displaystyle \frac{\partial z}{\partial x^j}}\right)\right]$$

(3)

$$S_n+S_w+S_a=1$$

(4)

where $x^i=\left(x,y,z\right)$ are the cartesian coordinates, ${\rho }_{\mathrm{n}},{\rho }_{\mathrm{w}},{\rho }_{\mathrm{a}}$ and ${\mu }_{\mathrm{n}},{\mu }_{\mathrm{w}},{\mu }_{\mathrm{a}}$ are the densities and viscosities of LNAPL, water, and air, respectively. $S_n,S_w,S_a,$ are the saturations of nonaqueous, water, and air, respectively, and $k_{rn},k_{rw},k_{ra}$ are the relative permeabilities that depend on the saturations with $k^{ij}$ the absolute permeability tensor $\left[L^2\right]$. The gravitational acceleration is represented by $g$, while $z$ is the depth, and $\varphi$ is the porosity of the porous medium. In a three-phase fluid flow, two capillary pressures are independent: the third can be written as a function of the other two. In our formulation, we chose $p_{can}$ and $p_{caw},$ the capillary pressures of air–nonaqueous and air–water, respectively. They are also a function of the various saturations. The rock compressibility as a function of the porosity and pressure is given by $c_R={\displaystyle \frac{1}{\varphi }}{\displaystyle \frac{\partial \varphi }{\partial \mathrm{p}}}$. From here the porosity can be calculated from a Taylor expansion as up to order one in $c_R:$

$$\varphi ={\varphi }_0\left[1+c_R\left(p-p_0\right)\right],$$

(5)

where ${\varphi }_0$ is the porosity at $p_0$ (the atmospheric pressure), and $p$ is the pressure that will be associated with $p_a$.

Let us define ${\sigma }_{\mathrm{n}}=S_n\varphi ,$ ${\sigma }_{\mathrm{w}}=S_w\varphi ,$ and ${\sigma }_{\mathrm{a}}=S_a\varphi$, that are the volume concentration in soil, for each phase. Using Equations (4) and (5) and assumed constant density–viscosity for each phase, the system (1)–(4) can be written as [35,36],

$${\displaystyle \frac{\partial {\sigma }_n}{\partial t}}+{\displaystyle \frac{\partial {\sigma }_w}{\partial t}}+{\displaystyle \frac{\partial {\sigma }_a}{\partial t}}={\varphi }_0{c}_R{\displaystyle \frac{\partial p}{\partial t}} ,$$

(6)

and $\alpha =\left(n,w,a\right),$

$${\displaystyle \frac{\partial {\sigma }_{\left(\alpha \right)}}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x^i}}\left[F_{\left(\alpha \right)}^i\left(S_w,S_n,S_a,p\right)\right]+{\displaystyle \frac{\partial }{\partial x^i}}\left[Q_{\left(\alpha \right)}^i\left(S_w,S_n,S_a,p\right)\right] ,$$

(7)

where $F$ does not depend on the spatial derivative of the saturation (proportional to gravity and pressure), while $Q$ depends on the spatial derivative of the saturation (proportional to the capillary pressures). The variables are in terms of the variables $S_n,S_w,S_a,$ and $p_a$, and the functional form of the relative permeabilities and capillary pressures.

The CactusHydro numerical code solves this governing equation using a high-resolution shock-capturing (HRSC) method that accurately follows sharp discontinuities and temporal dynamics [40,41,42]. CactusHydro is built on the Cactus computational toolkit [50,51,52], an open-source software framework for developing HPC numerical codes and utilizes a Cartesian mesh for evolving data using Carpet [53,54]. CactusHydro considers the migration of leaked LNAPL as immiscible and does not account for its effects on dissolution, biodegradation, etc.

The functional form for the relative permeabilities for three-phase fluid flow is extended from the two-phase expressions [55] and is given by

$$k_{rw}=S_{ew}^{1/2}{\left[1-{\left(1-S_{ew}^{1/m}\right)}^m\right]}^2$$

(8)

$$k_{ra}={\left(1-S_{et}\right)}^{1/2}{\left(1-S_{et}^{1/m}\right)}^{2m}$$

(9)

$$k_{rn}={\left(S_{et}-S_{ew}\right)}^{1/2}{\left[{\left(1-S_{ew}^{1/m}\right)}^m-{\left(1-S_{et}^{1/m}\right)}^m\right]}^2$$

(10)

where $S_{et}$ is the total effective liquid saturation, $S_{et}={\displaystyle \frac{S_w+S_n-S_{wir}}{1-S_{wir}}}$, and $S_{wir}$ is the irreducible wetting phase saturation. For the capillary pressures, we use the van Genuchten model [56] where the effective saturation is given by $S_e={\left[1+{\left(\alpha p_c\right)}^n\right]}^{\left(1-{\displaystyle \frac{1}{n}}\right)}$ and $\alpha ,\mathrm{n}$ are model parameters. From here, get the value of $p_c$: $p_c=-p_{c0}{\left(1-S_e^{1/m}\right)}^{1-m},$ where $m=\left(1-{\displaystyle \frac{1}{n}}\right),$ and $p_{c0}={\alpha }^{-1}$ is the capillary pressure at $S_e=0.$

2.3. Hydrogeological and Hydrocarbon Features

Nine new numerical simulations were performed using diesel oil as the contaminant spill. Its density is ${\rho }_n=830 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, thus lower than that of water, being an LNAPL, and its viscosity is ${\mu }_n=3.61\times {10}^{-3 }\mathrm{k}\mathrm{g}/\left(\mathrm{m}\mathrm{s}\right).$ See

Table 1. Definitions of the parameters used in the numerical simulations of diesel oil spill from an oil pipeline.

Parameter	Symbol	Value
Absolute permeability	$k$	$2.059\times {10}^{-11} {\mathrm{m}}^2\hspace{0.17em}$
Absolute permeability (bottom layer) 1	$k_b$	$2.059\times {10}^{-15} {\mathrm{m}}^2\hspace{0.17em}$
Porosity	${\varphi }_0$	0.43
Rock compressibility	$c_R$	$4.35\times {10}^{-7} \mathrm{P}{\mathrm{a}}^{-1}$
Diesel oil density	${\rho }_{\mathrm{n}}$	$830 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
Diesel oil viscosity	${\mu }_{\mathrm{n}}$	$3.61\times {10}^{-3} \mathrm{k}\mathrm{g}/\left(\mathrm{m}\mathrm{s}\right)$
Water density	${\rho }_{\mathrm{w}}$	${10}^3 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
Water viscosity	${\mu }_{\mathrm{w}}$	${10}^{-3} \mathrm{k}\mathrm{g}/\left(\mathrm{m}\mathrm{s}\right)\hspace{0.17em}$
Air density	${\rho }_{\mathrm{a}}$	$1.225 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
Air viscosity	${\mu }_{\mathrm{a}}$	$1.8\times {10}^{-5} \mathrm{k}\mathrm{g}/\left(\mathrm{m}\mathrm{s}\right)$
van Genuchten	$\left(n,m\right)$	$\left(2.68,1-{\displaystyle \frac{1}{2.68}}\right)$
Irreducible wetting phase saturation	$S_{wir}$	$0.045$
Superficial tension air-water	${\sigma }_{\mathrm{a}\mathrm{w}}$	$6.5\times {10}^{-2} \mathrm{N}/\mathrm{m}$
Superficial tension nonaqueous-water	${\sigma }_{\mathrm{n}\mathrm{w}}$	$3.0\times {10}^{-2} \mathrm{N}/\mathrm{m}$
Capillary pressure air-water at zero saturation	$p_{caw0}$	$676.55 \mathrm{P}\mathrm{a}$
Capillary pressure air-nonaqueous at zero saturation	$p_{can0}$	$374.78 \mathrm{P}\mathrm{a}$

¹ In this study, the influence of the impermeable bottom layer was taken into account. However, even under a high pipeline pressure of 20 atm, the LNAPL did not reach the bottom layer. Consequently, parameters associated with the bottom layer do not affect the simulation results under the present scenario.

for details. The porosity ${\varphi }_0$, at the reference pressure $p_0$(the atmospheric), is 0.43, compatible with ‘sand’ [57,58]. The rock compressibility $c_R$ for ‘sand’ is taken from [59]. The van Genuchten parameter, $\alpha =14.5 {\mathrm{m}}^{-1}$, is compatible for ‘clean sand’, ‘silty sand’, and the irreducible wetting saturation 0.045 is [58].

The hydraulic conductivity, $K=6.8\times {10}^{-5} \mathrm{m}/\mathrm{s}$, is considered constant and isotropic since that was the value obtained from the area of study. Using the expression, $k={\displaystyle \frac{K{\mu }_w}{{\rho }_{w}g}}$, we get, $k=2.059\times {10}^{-11} {\mathrm{m}}^2$. Using the values for the superficial tensions ${\sigma }_{nw}=3.0\times {10}^{-2} \mathrm{N}/\mathrm{m}$ and ${\sigma }_{aw}=6.5\times {10}^{-2} \mathrm{N}/\mathrm{m}$ we get, ${\beta }_{nw}={\displaystyle \frac{{\sigma }_{aw}}{{\sigma }_{nw}}}={\displaystyle \frac{6.5\times {10}^{-2} \mathrm{N}/\mathrm{m}}{3.0\times {10}^{-2} \mathrm{N}/\mathrm{m}}}=2.17,$ and $p_{cnw}\left(S_w\right)={\displaystyle \frac{p_{caw}}{{\beta }_{nw}}}=311.77 \mathrm{P}\mathrm{a}.$ From here we get $p_{can}=p_{caw}-p_{cnw}=676.55-311.77=374.68 \mathrm{P}\mathrm{a},$ where $p_{caw}=676.55 \mathrm{P}\mathrm{a}.$

2.4. Initial and Boundary Conditions of the LNAPL Migration into the Variably Saturated Zone

We consider an LNAPL spill, of density given in Table 1, inside an unsaturated zone. The contaminant is located at $\left(x,y,x\right)=\left(0,0,2\right) \mathrm{m},$ from a groundwater table surface that crosses the position $\left(x,y,x\right)=\left(0,0,0\right).$ The LNAPL is released at a fixed maximum pressure of 20 atm, which is a real value reported in the site [44] (while the unsaturated zone is at the atmospheric pressure) for a fixed time of 3600 s. The remaining unsaturated zone is at the atmospheric pressure. The initial saturation of the LNAPL is set to one, $S_n\left(0,0,2,t=0\right)=1$.

The grid geometry corresponds to a parallelepiped with a variably saturated zone for the migration of three-phase fluid flow (water + diesel oil + air) and a dimension of $160 \mathrm{m} \times 80 \mathrm{m}\times 19 \mathrm{m}$. A spatial grid resolution is dx = dy = dz =$0.50 \mathrm{m}$, and a time step resolution of $dt = 0.025 \mathrm{s}$. The extension of the grid is, $160 \mathrm{m}$ long from $x=\left[-100,+60\right] \mathrm{m}$, $80 \mathrm{m}$ wide from $y=\left[-40,+40\right] \mathrm{m}$, and $19 \mathrm{m}$ depth from $z=\left[+3,-16\right] \mathrm{m}.$ The dimension of the grid has been chosen large enough to avoid the effects of boundary conditions, specifically the finite grid size, on the dynamics of contaminant migration.

All boundary conditions are no-flow, except on top of the parallelepiped in the infiltration zone. The groundwater flow is directed from the right to the left and has a hydraulic gradient fixed to 0.04. The variably saturated zone has a constant absolute permeability of $2.059\times {10}^{-11} {\mathrm{m}}^2.$ The parallelepiped’s bottom part will behave as ‘impermeable’ with an absolute permeability of $2.059\times {10}^{-15} {\mathrm{m}}^2$, i.e., four orders of magnitud smaller than the rest of the system (see Table 1). The bottom layer elevation is $z=\left[-15,-16\right] \mathrm{m}.$

This study presents nine new numerical simulation results of diesel oil migration in a variably saturated zone, representing an emergency where an onshore oil pipeline breaks and the contaminant spills out for 3600 s. After that time, the system is closed, and the contaminant left continues its migration through the variably saturated zone. We analyze the following scenarios:

• Dry soil and pressure inside the oil pipeline equal to 1 atm = 101,325 Pa, 10 atm = 1,013,250 Pa, and 20 atm = 2,026,500 Pa, respectively;
• Unsaturated zone with $S_{w }=0.20$ and pressure inside the oil pipeline equal to 1, 10, and 20 atm, respectively;
• Unsaturated zone with $S_w =0.50$ and pressure inside the oil pipeline equal to 1, 10, and 20 atm, respectively.

The choice of the nine scenarios has a mapping to real-world analogs, e.g., dry season rupture (when the saturation is zero), rainfall-induced saturation (when the water saturation is either 0.20 or 0.50), different pipeline failure pressures. The boundary conditions in the case of water saturation different from zero were maintained at the upper boundary over all time, simulating a continuous percolation of water from the surface.

Table 2. Names assigned to each of the nine numerical simulations performed in this work.

Pressure Outflow	${\mathit{S}}_{\mathit{w}}=0$ (Dry Soil)	${\mathit{S}}_{\mathit{w}}=0.20$	${\mathit{S}}_{\mathit{w}}=0.50$
2,026,500 Pa	Sw = 0_20atm	Sw = 0.20_20atm	Sw = 0.50_20atm
1,013,250 Pa	Sw = 0_10atm	Sw = 0.20_10atm	Sw = 0.50_10atm
101,325 Pa	Sw = 0_atm	Sw = 0.20_atm	Sw = 0.50_atm

indicates the names of each of the nine numerical simulation scenarios. The objective is to investigate the dependence of the leaked diesel oil on pressure, the water saturation of the unsaturated zone in contaminant migration, and the percentage of contaminant mass distributed in the variably saturated zone. Here, we maintained a constant distance of two meters between the spill contaminant and the groundwater table. The dependence on the distance between the leaked oil contaminant and the groundwater table was investigated in a previous paper for dry soil [43].

3. Results and Discussions

3.1. Three Dimensional Transient Numerical Simulation Results

This section presents the results of the nine scenarios outlined in Table 2.

3.1.1. Pressure Outflow of 20 atm

First, it is considered the scenario Sw = 0_20atm, which represents a spill of diesel oil into the dry vadose zone from an onshore oil pipeline. See Table 2. The contaminant is ejected at a pressure of 20 atm for a fixed time of 3600 s. After that time, the system is closed, and the contaminant leak is stopped. The immiscible contaminant migrates into the unsaturated dry zone and toward the saturated zone (aquifer) due to the force of gravity until it reaches the groundwater table. Since this contaminant is an LNAPL, it remains in the groundwater table and does not migrate further into the saturated zone. See Figure 2. The left side column shows the 3D transient numerical results on the saturation contours, ${\sigma }_{\mathrm{n}}=S_n \varphi ,$ of three-phase immiscible fluid flow (water + diesel oil + air) at different times. The right-side column corresponds to 3D transient numerical results of the depth as a function of the diesel oil saturation (red line/points), water saturation (blue line/points), and air saturation (green line/points) at different times. The left side corresponds to the $z-x$ plane, and the right side corresponds to the $z-y$ plane. The pink rectangle is a zoom that shows the relevant part containing the contaminant.

Initially, the contaminant is positioned at $\left(x,y,z\right)=\left(0,0,2\right) \mathrm{m}$, as shown in Figure 2a, and it is situated two meters above the groundwater table. It is released for 3600 s. Its saturation is one. The porous medium is a variably saturated zone composed of an unsaturated zone (air–LNAPL for this dry situation and air–LNAPL–water when $S_w$ is different from zero) and the saturated zone (initially only water and eventually LNAPL). We consider a three-phase fluid flow model composed of water, air, and a fixed quantity of LNAPL for the transient numerical simulations. The blue arrows represent the direction of the groundwater flow.

Figure 2 (left side column) shows the saturation contours, ${\sigma }_{\mathrm{n}}=S_n \varphi$, of a three-phase immiscible fluid flow composed of water, LNAPL, and air in a porous medium (dry in this case) at different times. After 614 s, the diesel oil had already arrived at the groundwater table. See Figure 2b. After 3686 s, the contaminant enters the saturated zone due to the high pressure. See Figure 2c. The last panel shows the contaminant after 10 days and 17.3 h moving along the groundwater table in the left-hand direction due to a hydraulic gradient ($z -x$ plane). Notice that the zone around the oil pipeline contains some contaminants that were spread during the outflow. The rest migrates downward due to gravity. The part that is situated more deeply rises because the LNAPL remains near the groundwater table. Notice that a residual contaminant remains trapped in the unsaturated zone due to irreducible wetting phase saturation, as shown in Table 1 and Figure 2d. The right-hand side of the figure represents the $z-y$ plane, and it is symmetric around $y = 0$ since the hydraulic gradient does not affect it.

Figure 2 (right side column) shows the elevation as a function of the saturation contours of the three-phase fluid flow $S_{n,w,a}\left(\mathrm{z},\mathrm{t}\right)={\sigma }_{\mathrm{n},\mathrm{w},\mathrm{a}}/\varphi$ at different times. The red line/points represent the diesel oil saturation $\left(S_n\right),$ the blue line/points represent the water saturation $\left(S_w\right),$ and the green line/points represent the air saturation $\left(S_a\right).$ Initially, at t = 0 s, Figure 2e, there is a shock front of immiscible diesel oil located at $\left(x,y,z\right)=\left(0,0,2\right)$ with a constant saturation $S_n=1.$ The red line abruptly drops to zero before and after this position, as the contaminant is initially situated on top of the grid (the same for all scenarios). Similarly, the air saturation (green line) differs from zero in the unsaturated zone, except where the contaminant is located, and then returns to zero in the saturated zone. The water saturation (blue line) is zero in the unsaturated zone (Sw = 0_20atm scenario has dry soil) and is one in the saturated zone, as it is filled with water.

After 614 s, as shown in Figure 2f, the shock front (red line) decreases due to the gravitational force and the pressure of the spilled oil pipeline. It has already reached the groundwater table. After 3686 s, as shown in Figure 2g, the contaminant (red line) entered the saturated zone and reached a depth of −2.5 m in the z elevation. See also Figure 2c. After 10.7 days, as shown in Figure 2h, the contaminant saturation decreases from its initial value of one to a value around 0.2. It remains trapped in the dry soil due to a nonzero value of the irreducible wetting phase saturation. On the other hand, its saturation increases around the groundwater table since the LNAPL tends to remain in the capillary fringe. Moreover, the part of the diesel oil that initially reached the saturated zone at −2.5 m, in Figure 2c, returns to the capillary fringe zone (now at an elevation of −1.0 m). The other blue and green lines are positioned accordingly, as the sum of all saturations is one at any time.

Figure 3 shows the saturation contours, ${\sigma }_{\mathrm{n}}=S_n \varphi ,$ for the scenario Sw = 0_20atm, of the previous Figure 2, but in the plane $y-x,$ and a zoom on the right-hand side, after 9.1 h. We picked this time to compare with all scenarios easily. The $y-x$ plane cross the point $\left(x,y,z\right)=\left(0,0,-1\right) \mathrm{m}$. Consequently, the immiscible diesel oil is shown only when it arrives at the groundwater table. Notice that the diesel oil (in all scenarios) is moving to the left due to a hydraulic gradient of 0.04. After 9.1 h, the contaminant reached the groundwater table with an approximately oval shape, ranging from −5 m to 5 m. The green colors represent the water saturation, as in Figure 2.

The effects on water saturation of the unsaturated zone are investigated and considered in the scenario Sw = 0.20_20atm, like the previous case, but with a saturation of $S_w=0.20$ in the unsaturated zone. See Figure 4. This situation can be imagined as a continuum of rain, partially filling the unsaturated zone with water. In this case, the initial and boundary conditions are such that at each point of the grid in the unsaturated zone, the water saturation is 0.20, and at later times, on top of the grid, this value is kept constant. In this way, the water saturation of the unsaturated zone remains constant at 0.20. That is why we observe the blue arrows inside the grid, corresponding to a vertical water flow downward in the unsaturated zone due to the gravitational force (in addition to the arrows going to the left in the saturated zone due to a hydraulic gradient). Figure 4 shows transient numerical results for (a) zero s, (b) 614 s, (c) 3686 s, (d) 2 days and 14.1 h.

Figure 4 (the right-hand column) shows the elevation as a function of the water, diesel oil, and air saturations of the left-hand column (scenario Sw = 0.20_20atm) at different times. Compared to the dry soil case (Figure 2), there are differences. Initially, at t = 0 s, Figure 4e, the contaminant whose saturation corresponds to the red line is located at $\left(x,y,z\right)=\left(0,0,2\right) \mathrm{m}$. The blue line representing water saturation is 0.20, and the green line is 0.80 in the unsaturated zone (and their sum is 1.0), except in the zone where the contaminant is located. Then, the blue line is 1.0 in the saturated zone, as in the previous case in Figure 2, since the saturated zone is filled with water.

After 614 s, as shown in Figure 4f, the contaminant has migrated downward due to gravitational force, and its saturation decreases in the unsaturated zone as it moves downward. Contextually, the water saturation increases in the same vadose zone. See also t = 3686 s, Figure 4g. After 2.6 days, as shown in Figure 4h, the contaminant has migrated downward and remains near the capillary fringe (and groundwater table). There is a residual saturation of the contaminant that remains trapped at depth. A contaminant saturation between 0.2 and 0.4 remains between the elevation [0, 2.5] m, as there is a residual wetting phase saturation (see Table 2). The rest of the contaminant is positioned in the capillary fringe, as diesel oil is an LNAPL.

Figure 5 shows the saturation contours, ${\sigma }_{\mathrm{n}}=S_n \varphi ,$ (scenario Sw = 0.20_20atm), of Figure 4, but in the plane $y-x,$ and a zoom on the right-hand side, after 9.1 h. The contaminant that arrived in the plane is similar to the one in Figure 3, although the saturation seems to be slightly lower.

Consider now the scenario Sw = 0.50_20atm, like the previous situation, but with a saturation of $S_w=0.50$ in the unsaturated zone. Figure 6 shows the transient 3D numerical results on the saturation contours, ${\sigma }_{\mathrm{n}}=S_n \varphi ,$ of three-phase immiscible fluid flow (water + diesel oil + air) at different times and the elevation as a function of the saturations. The green colors represent the water saturation. In this case, the green coloration in the unsaturated zone is more pronounced than in the previous cases. As before, there is the shock front of contamination at the initial time. Subsequently, the diesel oil reaches the groundwater table at 614 s, as shown in Figure 6b. Since the unsaturated zone is partially saturated with water, there is less contaminant outflow from the oil pipeline compared with previous cases. The contaminant goes downward due to the gravitational force and the pressure of the oil pipeline. The following sections will quantify the contaminant mass distribution in the saturated and unsaturated zones, as well as a comparison of all scenarios. In any case, after 2 days and 16 h, as is shown in Figure 6d, the contaminant is in the capillary fringe, and it spreads around this zone. Additionally, the groundwater table surface no longer crosses the point $\left(x,y,z\right)=\left(0,0,0\right) \mathrm{m}$, as the water in the unsaturated zone causes the elevation to increase.

Figure 6 (right side column) shows the elevation as a function of water, diesel oil, and air saturations (scenario Sw = 0.50_20atm) at different times. Basically, the difference with respect to the previous situation (Scenario Sw = 0.20_20atm) is that, at t = 0 s, the water saturation (blue line/points) is positioned at 0.50 in the unsaturated zone, except for the contaminant that has a saturation of 0.9 (and 0.1 for the water saturation at this point); see Figure 6e. After 614 s and 3866 s, as shown in Figure 6f and Figure 6g, respectively, the contaminant saturation is lower than in the previous dry soil scenario in Figure 2 since, as expected, part of the porous medium is filled with water. For this reason, the presence of water in the unsaturated zone makes contamination in the porous medium less effective. Despite this, the presence of water saturation in the unsaturated zone increases the velocity of the contaminant migration. See also Figure 4f,g. Comparing Figure 6h with Figure 4h, there is more contaminant distribution around the capillary fringe/groundwater table for the scenario Sw = 0.20_20atm than the scenario Sw = 0.50_20atm. Figure 7 shows the saturation contours, ${\sigma }_{\mathrm{n}}=S_n \varphi ,$ of Figure 6, but in the plane $y-x,$ and a zoom on the right-hand side after 9.1 h. The contaminant that arrived in the plane is similar to the one in Figure 5, although the saturation seems to be slightly lower, as expected.

3.1.2. Pressure Outflow of 10 atm

The scenario Sw = 0_10atm (dry soil) in Figure 8 is similar to the scenario Sw = 0_20atm in Figure 2, but with a pressure of 10 atm inside the oil pipeline instead of 20 atm. The contaminant migrates downward due to gravity and the outflow pressure of the oil pipeline. After t = 615 s, the contaminant has just arrived at the groundwater table. See Figure 8b,f. This case differs slightly from the one in Figure 2b,f, where pressure is higher. The effects on the oil pipeline pressure are more pronounced. Compare also Figure 8c,g with Figure 2c,g after 3686 s. Similarly, the saturation contours in the $y - x$ plane in Figure 9 do not substantially change compared to Figure 3, although the saturation seems to be lower and the oval shape a bit smaller. Thus, it is essential to quantify the distribution of contaminant mass remaining in the migration process within the porous medium to determine the roles of pressure and water saturation in contaminant migration.

Figure 10 and Figure 11, as well as Figure 12 and Figure 13, show the scenarios Sw = 0.20_10atm and Sw = 0.50_10atm at different times. They are similar to the previous case (dry soil) in Figure 8 and Figure 9, but with a water saturation of 0.20 and 0.50, respectively. After 614 s and 3686 s, the diesel oil had already reached the groundwater table in both scenarios (see Figure 10b,c,f,g). However, it first arrived for the scenario Sw = 0.50_10atm (Figure 12b,c,f,g). The presence of water saturation in the unsaturated zone slightly accelerates the arrival time of contaminants into the aquifer. On the other hand, the effects on the oil pipeline pressure are much stronger than those on the water saturation of the unsaturated zone. Also, there is a vertical water flow, since the water saturation of the unsaturated zone is now 0.20 (or 0.50). In the $y - x$ plane in Figure 11 and Figure 13 after 9.1 h, the scenario is essentially the same as the one in Figure 5 and Figure 7, where the outflow pressure is 20 atm, although, there is more contamination in the groundwater table due to the higher pressure.

3.1.3. Pressure Outflow of 1 atm

Here, we investigate the scenario in which the outflow pressure of the oil pipeline is 1 atm. Typically, the outflow pressure is significantly elevated; however, it is interesting to numerically investigate the fate of the migration and compare it with previous scenarios involving elevated pressure values. Firstly, pressure plays a crucial role in the migration of the contaminant. Compared to the earlier cases of 10 atm and 20 atm, and aside from the gravitational force, the migration is significantly slower here. See Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19. After 3686 s, the contaminant still has to arrive at the groundwater table, as shown in Figure 14c,g (scenario Sw = 0_atm), Figure 16c,g (scenario Sw = 0.20_atm), and Figure 18c,g (scenario Sw = 0.50_atm). Additionally, in the $y-x$ plane as shown in Figure 15, Figure 16 and Figure 17.

3.2. Quantitative Comparison of the Saturation Results for the Various Scenarios

The distribution in mass of the diesel oil in a variably saturated zone is quantitatively investigated through the saturation contours results of the diesel oil multiplied by the porosity, ${\sigma }_n=S_n\varphi$. These data are taken from the output of the numerical simulation results for the various scenarios presented in Figure 2, Figure 4, Figure 6, Figure 8, Figure 10, Figure 12, Figure 14, Figure 16 and Figure 18.

Table 3. Mass in Kg and contaminant percentage after 3600 s as a function of the elevation.

z Elevation (m.a.s.l.)	Mass in kg of LNAPL After 3600 s	Percentage of Trapped Contaminant (%)
Sw = 0_20atm (dry soil)
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$3.45$	$0.01$
$-1.75<z<-0.75$	$1347.53$	$3.66$
$-0.75<z<0.25$	$5842.19$	$15.85$
$0.25<z<1.25$	$\mathrm{11,771.02}$	$31.94$
$1.25<z<2.25$	$\mathrm{12,469.04}$	$33.83$
$2.25<z<3.75$	$5425.77$	$14.72$
Sw = 0.20_20atm
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$12.42$	$0.04$
$-1.75<z<-0.75$	$1685.74$	$4.89$
$-0.75<z<0.25$	$5990.61$	$17.38$
$0.25<z<1.25$	$\mathrm{10,639.89}$	$30.88$
$1.25<z<2.25$	$\mathrm{11,251.34}$	$32.65$
$2.25<z<3.75$	$4880.67$	$14.16$
Sw = 0.50_20atm
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$38.78$	$0.13$
$-1.75<z<-0.75$	$2068.82$	$6.72$
$-0.75<z<0.25$	$5894.94$	$19.15$
$0.25<z<1.25$	$8921.16$	$28.98$
$1.25<z<2.25$	$9721.97$	$31.58$
$2.25<z<3.75$	$4142.81$	$13.46$
Sw = 0_10atm (dry soil)
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$0.00$	$0.00$
$-1.75<z<-0.75$	$73.73$	$0.36$
$-0.75<z<0.25$	$2441.22$	$11.84$
$0.25<z<1.25$	$6679.33$	$32.40$
$1.25<z<2.25$	$7888.66$	$38.27$
$2.25<z<3.75$	$3529.36$	$17.12$
Sw = 0.20_10atm
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$0.00$	$0.00$
$-1.75<z<-0.75$	$138.00$	$0.72$
$-0.75<z<0.25$	$2622.88$	$13.66$
$0.25<z<1.25$	$6187.01$	$32.21$
$1.25<z<2.25$	$7108.42$	$37.01$
$2.25<z<3.75$	$3150.03$	$16.40$
Sw = 0.50_10atm
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$0.00$	$0.00$
$-1.75<z<-0.75$	$228.32$	$1.34$
$-0.75<z<0.25$	$2620.97$	$15.40$
$0.25<z<1.25$	$5282.26$	$31.03$
$1.25<z<2.25$	$6208.22$	$36.47$
$2.25<z<3.75$	$2684.50$	$15.77$
Sw = 0_atm (dry soil)
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$0.00$	$0.00$
$-1.75<z<-0.75$	$0.00$	$0.00$
$-0.75<z<0.25$	$0.00$	$0.00$
$0.25<z<1.25$	$0.30$	$0.21$
$1.25<z<2.25$	$87.19$	$61.92$
$2.25<z<3.75$	$53.33$	$37.87$
Sw = 0.20_atm
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$0.00$	$0.00$
$-1.75<z<-0.75$	$0.00$	$0.00$
$-0.75<z<0.25$	$0.00$	$0.00$
$0.25<z<1.25$	$1.26$	$0.90$
$1.25<z<2.25$	$85.68$	$61.19$
$2.25<z<3.75$	$53.08$	$37.91$
Sw = 0.50_atm
$-3.75<z<-2.75$	$0.00$	$0.00$
$-2.75<z<-1.75$	$0.00$	$0.00$
$-1.75<z<-0.75$	$0.00$	$0.00$
$-0.75<z<0.25$	$0.00$	$0.00$
$0.25<z<1.25$	$3.63$	$2.65$
$1.25<z<2.25$	$81.43$	$59.38$
$2.25<z<3.75$	$52.08$	$37.98$

presents the contaminant mass in kilograms expelled for 3600 s, and the trapped contaminant percentage as a function of the elevation in the various scenarios.

The contaminant mass is calculated using the expression:

$$M_n={\rho }_{\mathrm{n}}\sum _{ijk}{ \sigma }_{nijk}\times \left(dx dy dz\right)$$

(11)

where ${\sigma }_{nijk}$ is the contaminant saturation multiplied by the porosity in the cell $i,j,k,$ and $\varphi$ is the porosity value given in Equation (5), while ($dx dy dz$) are the side lengths of cells $i,j,k.$ Table 3 shows the contaminant mass distribution in different zones defined in the first column, while the third zone shows the results in percentage.

Figure 20 presents the data from Table 3, which summarizes all the results investigated in this paper. The elevation as a function of the leaked contaminant mass in kg for a fixed time of 3600 s and the various scenarios: (a) 10 atm and 20 atm; (b) 1 atm. For each pressure value, it also shows the water saturation of the unsaturated zone.

The pressure is the parameter that most significantly affects the contaminant migration and the time of arrival at the groundwater table. Indeed, in Figure 20a, the contaminant spill is directly proportional to its pressure value. The higher the pressure value, the greater the quantity of released contaminant (at a fixed time). In fact, for one atmosphere of pressure in the oil pipeline (Figure 20b), the quantity of released contaminant is significantly lower than in the previous cases.

Additionally, the water saturation of the unsaturated zone affects contaminant migration and the arrival time at the groundwater table, albeit to a lesser extent than the oil pipeline pressure. Indeed, the arrival time at the groundwater table increases as the water saturation decreases. See Figure 20a. For the case of 20 atmospheres, the dry soil case is the scenario in which the spilled contaminant is greater than the $S_w=0.20$ and $S_w=0.50$. The same applies to the case of 10 atmospheres. However, at the same time, the case $S_w=0.50$ is the one that contains more mass accumulation around the groundwater table (see the values at around $z=\left[-1,0\right]$), which means that the contaminant arrives before the other cases with $S_w=0.50$, and dry soil. A similar situation happens for the scenario with 10 atmospheres (Figure 20a).

For the case of one atmosphere (Figure 20b), the situation corroborates the previous cases (20 atm and 10 atm). Indeed, although the spilled contaminant is proportional to the pressure, the water saturation of the unsaturated zone has a milder influence on the fate of the contamination. The more saturated the unsaturated zone is, the more contaminants it accumulates at the groundwater table/aquifer. See the mass values at the interval of elevation of $z=\left[-1,0\right].$ If it has rained recently in the area of interest, then this fact can reduce the quantity of contaminant released; however, the contaminant that is released arrives more easily and spreads into the aquifer.

3.3. Validation of the Numerical Results

The numerical simulations were validated through convergence tests, which involved running the same numerical code at different resolutions. Figure 21 shows an example of numerical results on the saturation contours for the diesel oil pipeline spill using a grid resolution of $dx=dy=dz=0.50 \mathrm{m}$ (left-hand side) or $dx=dy=dz=0.25 \mathrm{m}$ (right-hand side). As can be seen, both column shows analogous results when using two different resolutions within a 5% error.

4. Conclusions

The spilled hydrocarbons released from oil pipeline accidents can result in long-term environmental contamination and significant damage to habitats. For this reason, it is fundamental to evaluate actions in response to emergency management and groundwater integrity. In this regard, understanding the trajectories and their influence on various parameters related to contaminant migration through numerical simulations can help in developing a rapid remediation strategy.

This paper develops a numerical model using the CactusHydro code and the HRSC flux conservative method recently introduced in [35,36], which analyzes nine new different emergency scenarios, with respect to a previous work [43], that encompass conditions such as dry season rupture, rainfall-induced saturation, and varying pipeline failure pressures. The migration of LNAPL contaminants is investigated as a function of diesel oil pipeline pressure and water saturation in the unsaturated zone. We investigated three different values of oil pipeline pressure and three different values of water saturation. We follow with the high-accuracy formation of shock fronts of the advective part of the migration. We also quantified the mass distribution of the contaminant along the variably saturated zone expelled during the emergency for each scenario. The results indicate that oil pipeline pressure is the parameter that most influences contaminant migration. The aquifer contamination strongly depends on the pressure outflow in the vertical flow. For a fixed value of the outflow pressure, as the water saturation increases, the quantity of contaminant expelled decreases; however, at the same time, the contamination speed increases, allowing the pollution to reach more extensive areas.

The results indicate that CactusHydro can accurately follow the formation of the shock front of the advective part of the migration, which is a fundamental aspect to consider in the remediation strategy. For the moment, we consider only immiscible three-phase fluid flow and no dissolution or biodegradation. This will be the focus of a future investigation.