DOI: 10.3303/CET2188080 
 

 
 

 
 

 

 
 
 

 
 
 

 

 

 
 
 

 
 
 

 
 
 

 
 

 
 
 

 
 
 

 
 
 

 
 

 

 
 

Paper Received: 17 May 2021; Revised: 13 August 2021; Accepted: 2 October 2021 
Please cite this article as: Agranat V., Romanenko S.V., Perminov V., 2021, Mathematical Modelling of River Pollution as a Result of Pipeline 
Damage, Chemical Engineering Transactions, 88, 481-486  DOI:10.3303/CET2188080 

CHEMICAL ENGINEERING TRANSACTIONS

VOL. 88, 2021 

A publication of

The Italian Association
of Chemical Engineering
Online at www.cetjournal.it 

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Copyright © 2021, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-86-0; ISSN 2283-9216

Mathematical Modelling of River Pollution as a Result of
Pipeline Damage

Vladimir Agranata, Sergey Romanenkob,c, Valeriy Perminovb,*
a ACFDA, 81 Rejane Crescent, Thornhill, Ontario, L4J 5A5, Canada  
b Tomsk Polytechnic University, Lenin avenue, 30, Tomsk, 634050, Russia 
c Siberian State Medical University, Moskovsky trakt 2, Tomsk, 634050, Russia 

perminov@tpu.ru

A mathematical model of the process of pollution of the aquatic environment with oil products at the rupture of
an oil pipeline crossing a river bed is presented. To describe this process, Reynolds equations are used for
turbulent flow. The discrete analogue is obtained using the finite volume method. As a result of the numerical
solution, the influence of the river parameters and the characteristics of the source of pollution on the process
of distribution of petroleum products in the aquatic environment were studied. The results of mathematical
modelling of oil spills can be used in the development of preventive measures to detect and prevent the
spread of pollution.

1. Introduction

The first studies simulated the spread of petroleum products in calm waters. The oil slick was considered to be
round. In this case, only an increase in its diameter was considered (Buckmaster, 1973). These problems
were solved in one-dimensional or axisymmetric formulations. Later, models were developed that more
realistically simulated the processes of the spread of oil products in the aquatic environment. They took into
account the processes caused not only by the action of wind and sea currents, but also evaporation,
dissolution, emulsification, etc. The literature contains data on mathematical modelling of water pollution by oil
products, but they mainly relate to the study of pollution in the seas (Li et al., 2007). The study of Guandalini et
al. (2017) described the development of a methodology to evaluate the dynamics related to the dispersion of
hydrocarbons in the sea. In contrast to previous paper using two-dimensional approach (Issakhov et al.,
2021), the spatial case is considered in this paper. It should be noted that most of the papers on the pollution
of the aquatic environment with oil products is mainly devoted to either modeling the flow of oil in pipes and
assessing damage in case of rupture of oil pipelines (Sun, 2012), or accidental pollution of the seas from
ships. In paper (Xu et.al. (2019)), an attempt is made to study the process of oil products outflow from the
back surface of an underwater blunt body in the sea in a two-dimensional setting. Unfortunately, this paper
does not contain data on the further spread of oil products in the aquatic environment. In the presented paper,
we consider a section of a river with a given flow rate, at the bottom of which a pipeline is located in the
underwater crossing system. The situation is considered when damage to the pipeline occurs and oil products
enter the open water body. The processes occurring in the water when petroleum products get into it are
influenced by: wind speed and direction, flow velocity and river geometry, as well as damage characteristics
and power output. Oil is able to evaporate, dissolve, biodegrade, photo- and thermochemically decompose,
emulsify with water, settle at the bottom and float to the surface, adsorb and absorb, envelop suspended
particles, and also interact with biological organisms. The study area is considered to be the part of the river
near the pipeline rupture.

2. Mathematical and Physical Setting

In this problem, a section of a river with a given flow rate is considered, at the bottom of which there is a
pipeline in the underwater crossing system. The situation is considered when the pipeline is damaged and oil

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products enter the open water body. The processes occurring in water when oil products get into it are
influenced by: wind speed and direction, flow speed and geometry of the river, as well as the characteristics of
damage and the power of the release. Oil can evaporate, dissolve, biodegrade, photo- and thermochemically,
emulsify with water, settle at the bottom and float to the surface, adsorb and be absorbed, envelop suspended
particles, and also interact with biological organisms. A part of the river near the rupture of the oil pipeline is
considered as a research area. At the lower boundary of the computational domain, the parameters of the
pollution source formed as a result of damage to the oil pipeline are set (Figure 1). The direction of the flow of
the river coincides with the direction of the x1 axis, and the flow rate from the intended opening of the pipeline
with the direction of the x3 axis. The purpose of this paper is to obtain the distribution of fields of concentration
of petroleum products flowing from the pipeline, temperature and velocity, as well as to determine the
distribution area of pollutants to predict this emergency situation and its subsequent elimination. A section of
the river was considered in the form of a parallelepiped of size (6m × 4m × 2m). The coordinates and surface
areas of the source of pollution at the bottom of the river, as well as the rate of release of petroleum products
from the hole were specified.

Figure 1: The calculation domain. 

Table 1: Dependent variables, effective exchange coefficients and source terms in equation (1) 

Conservation of Φ Гф Sф
mass 1 0 0

xi - momentum ui  tµ µ  i
i

p
g

x



 


Enthalpy h
Pr

Pr
t

t



 0

Mass of α – species cα
t

t
Sc Sc


 0

Turbulent kinetic energy k
t

k





 ( )

k k
P W    

Dissipation rate of turbulent
kinetic energy

ε
t









1 2 3
( )

k k RNG
C P C C W R

k
  


     

The density of oil is ρ = 800 kg/m3. For transportation of oil, it is usually heated in specially designated points.
In the first case, the temperature of the oil at the time of its flow is 5 °C, in the second – 15 °C. The flow rate of
oil from the pipeline is 0.6 m/s. The water temperature is 10 °C and the flow velocity of the river: in the first
case - 0.5 m/s, and in the second - 1 m/s. Mathematically, this problem is reduced to solving the Reynolds
equations for turbulent flow, as well as the equations of energy and concentration of the pollutant. According
to Patankar (1981), the generalized differential equation for the desired dependent variables (components of
the velocity vector, components of concentrations, energy, etc.) will have the following form:

  i
i i

u S
t x x
 

 

   
     

   

 (1)

where t is time; xi - spatial coordinate (i = 1, 2, 3); ρ is the density of the liquid; ui is the velocity component in
the xi direction, Φ is the generalized dependent variable, ΓΦ is the mass transfer coefficient, and SΦ is the

482



source term. Specific values of the listed quantities are given in Table 1. The density of the liquid is calculated
from the equation of state for mixtures of liquid: p - pressure, T - absolute temperature, R - universal gas
constant; сα - mass fraction of α; α - component of the mixture (α =1 – oil, 2 - water). Here k is the turbulent
kinetic energy; ε is the dissipation rate of turbulent kinetic energy; µ and µt are the dynamic molecular and
turbulent viscosities; Pr, Sc, Prt and Sct are the molecular and turbulent Prandtl and Schmidt numbers; Ϭk, Ϭε,, 
Cµ, Cε1, Cε2, Cε3 are the empirical constants of turbulent model; gi is the gravity acceleration component (𝑔 =
(0,0, −g); ui is the velocity vector component; Pk, Wk are the turbulent production terms; RRNG is an additional
term proposed in the RNG k-ε model.

2

1

c
p RT

M



 




  (2)

3. Results Calculation and Analysis

The boundary-value problem is solved numerically using finite volume method (Patankar, 1981) and software
PHOENICS. As a result of the numerical solution of the problem posed, spatial distributions of the fields of
velocity, temperature, and concentration of petroleum products are obtained. In Figure 2 shows the
concentration distribution of pollution in the river at a flow rate of 0.5 m/s in the section of the vertical plane
x1Ox3 at different times after the start of oil release from the hole in the pipeline. The highest concentration
values are located directly next to the place of oil outflow from the pipe. As the pollutant is transferred by the
flow of an oil spill, its concentration decreases. Figure 3 shows the distribution of petroleum products on the
water surface in the x1Ox2 plane. As the stain spreads in the water, the oil stain on the surface takes the form
of a horseshoe, due to the fact that the flow of the pollutant affects the flow of the river

Figure: 2: C1 distribution at different times (in the x1x3 plane) 

Figure: 3: The distribution of c1 on the surface of the water at different moments of time 

Figure 4 clearly shows the deceleration of the current velocity near the place of emission of pollutants, both to
the left and to the right of the source. In this case, with a river depth of 2 m and a river flow velocity of 0.5 m /
s, oil appears on the surface of the reservoir 5 seconds after the start of the emission source. The distance
from the lifting point to the oil pipeline is 3 meters. When the speed of the river flow doubles, the oil slick
released during the emergency release also appears on the water surface at a greater distance from the
source of oil release from the pipeline. Figure 5 shows the calculation results for a river flow rate of 1 m/s.

483



Figure 4: The distribution of the velocity field at different moments of time 

Oil emulsions will be carried by the current of the river to a more distant distance and reach the surface in a
greater amount of time. According to calculations, it will be possible to see the spots carried away on the
surface of the river after 12 m from the place where the pipeline is located.

Figure 5: The distribution of the concentration of C1 at a flow rate of 1 m/s 

The volume of the leaked oil and its negative impact on the environment can be very significant even with a
small hole, if it has been a long time since the fistula was formed. Nevertheless, with an increase in the size of
the hole that appeared when the oil pipeline is damaged, the volume of oil release into the aquatic
environment, of course, increases. Figures 6 a) and b) show the dynamics of the release of oil products for
various sizes of holes in oil pipelines: a) S1 = 0.0225 m2 and b) S2 = 0.09 m2. As can be seen in Figure 6,
already 3 seconds after the start of the blowout near the source of the outflow, the amount of oil flowing out
increased, as evidenced by the distribution of its concentration.

Figure 6: The distribution of the concentration of c1 by increasing the opening of the fistula 

This model makes it possible to track the spatial distribution of oil in the river bed over time for various
conditions associated with both the nature of the accident and the properties and parameters of the river (flow
velocity, configuration, depth, water temperature, etc.). For example, Figures 7 and 8 show the spatial
distributions of the concentration of oil products in the riverbed at different points in time. 

484



Figure 7: The distribution of c1 at the moment t=8 sec. 

Figure 8: The distribution of c1 at the moment t=18 sec. 

The model developed in this paper allows one to obtain the spatial distributions of the sought functions, in
particular, the velocity fields and pollutants. Thus, Figures 9 and 10 show the distribution of contamination
spread at different points in time and the vector velocity field in the river bed at the corresponding time points
of 8 and 18 seconds on the surface of the river.

Figure 9: The distribution of c1 and vector field of velocity at the moment t=8 sec. 

485



Figure 10: The distribution of c1 and vector field of velocity at the moment t=8 sec.

4. Conclusion

Thus, this paper presents a mathematical model of the distribution of petroleum products in the aquatic
environment when the pipeline ruptures. Using the PHOENICS software, spatial distributions of pollution in the
river bed were obtained depending on various parameters. The influence of the speed of the river flow, its
depth, as well as the size of the pollution source has been studied. The developed mathematical model and
the created program make it possible to obtain the spatial distribution of pollutants in the river bed. The results
of calculations using this model can be used to determine the location of sensors for recording pollution of the
aquatic environment, as well as the dynamics of the spread of oil products over time in the river bed. This will
ensure the effectiveness of the early monitoring system.

Nomenclature 

Cε1, Cε2 , Cε3 - empirical constants
cα - mass concentrations of α - component
g – acceleration of gravity, m/s2
k - - turbulent kinetic energy, J
Mα - molecular weight of α - component
p - pressure, n/m2
Pk - the turbulent production term 
Pr – Prandtl number
R – gas constant, J/(mol·K)
RRNG - additional term in the RNG k-ε model
Sc - Schmidt number

S1, S2 – hole area, m2
t – time, s
T – temperature, K
ui - velocity component in xi direction, m/s
Wk - the turbulent production term
xi – cartesian coordinates, m
ГΦ - coefficient of turbulent transfer
ε - dissipation turbulent kinetic energy, m2/s2 
µt - coefficient of turbulent viscosity, kg/(m·s)
ρ - density of the liquid, kg/m3
Ϭk, Ϭε - empirical constants -

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