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American Journal of  
Environment and Climate (AJEC)

Modeling of  Hydrodynamic Dispersion in the Sidi Kirayr Locality’s 
Coastal Unconfined Aquifer West of  Alexandria, Egypt

Mohamed M. M. Amin1, Medhat M. H. ElZahar1*

Volume 2 Issue 1, Year 2023
ISSN: 2832-403X (Online)

DOI: https://doi.org/10.54536/ajec.v2i1.1188
https://journals.e-palli.com/home/index.php/ajec

Article Information ABSTRACT

Received: January 08, 2023
Accepted: March 02, 2023
Published: March 11, 2023

The finite difference method was used in this study to solve the two-dimensional 
hydrodynamic dispersion equation for a typical coastal unconfined aquifer. A mathematical 
model is prepared to solve this dispersion problem, which is then modeled and analyzed 
over time. The salinity profiles for the aquifer transition zone in the Sidi Kirayr locality are 
determined using the finite difference method. The computed salinity profiles are compared 
to those obtained from an actual well drilled in the area and used to determine electric 
conductivity. Profile correlation reveals a roughly parallel trend along the tested depth 
interval. Additionally, these results correlate well with the chemical analysis of  water from 
the same well. The current work promotes the application of  the solution methodology in 
comparable coastal regions in the absence of  dispersion measures for the analyzed aquifer. 
This innovative approach can be used to forecast groundwater quality in comparable 
coastal regions. It can also help in selecting the ideal sites for man-made wells that pump 
groundwater.

Keywords
Coastal Area, Dispersion, 
Dispersion Co-efficient, 
Hydrodynamic Dispersion, 
Salt Concentration

1 Department of  Civil Engineering, Faculty of  Engineering, Port Said University, Port Fouad 42523, Port Said, Egypt 
* Corresponding author’s e-mail: medhat.alzahar@eng.psu.edu.eg

INTRODUCTION
Many of  us think of  lakes, rivers, and streams as sources 
of  water, in other words, surface water. However, 
approximately 97% of  the world’s usable freshwater 
is groundwater. According to the UN, underground 
storage of  10 million cubic kilometers of  water exists. 
According to the United States Geological Survey, there 
are approximately 4.2 million cubic kilometers of  water 
within 0.8 kilometers of  the earth’s surface. Because 
of  the distance between the northwestern coastal line 
and the Nile River, Egypt’s northwestern coastal zone 
requires novel approaches to water conservation. The 
study of  the quality of  groundwater and its suitability for 
human use is an important environmental issue. Because 
of  the long distance between these areas and the Nile 
River, Egypt’s main source of  fresh water, studying the 
quality of  groundwater in coastal areas is an important 
issue. This study used the finite difference method 
and a computer program to solve the two-dimensional 
hydrodynamic dispersion equation with time for a typical 
coastal unconfined aquifer. 
The finite difference method is used to detect salinity 
profiles for the aquifer’s transition zone in the Sidi Kirayr 
locality. The computed salinity profiles are compared to 
those obtained from an actual well drilled in the area and 
used to measure electric conductivity. The correlation 
of  profiles shows a nearly parallel trend along the tested 
depth interval. These findings are also consistent with the 
chemical analysis of  water from the same well, indicating 
good agreement. The study promotes the application of  
the mathematical approach in comparable coastal regions 
in the absence of  dispersion measurements for the studied 
aquifer. This can be used to forecast the water quality in 
nearby coastal regions. It can also help in selecting the 
ideal sites for man-made wells that pump groundwater.

When a freshwater coastal aquifer is discharged into the 
sea, fresh and saline water meet, and a transition zone 
is formed between the two types of  water. Although 
the freshwater is lighter than the saltwater, a mixing of  
saltwater and freshwater occurs. However, the interface 
formed between the fresh and the underlying saltwater 
is not sharp but is represented by a mixing zone. The 
salinity of  this zone varies from that of  freshwater to that 
of  seawater downwards.
Hydrodynamic dispersion in unconfined aquifers has been 
a problem in groundwater exploitation and utilization in 
many coastal aquifers in Egypt’s northwestern coastal 
zone, such as the coastal unconfined freshwater aquifer 
in the Sidi Kirayr locality, 32 kilometers west of  the city 
of  Alexandria, Figure 1. However, the hydrodynamic 
dispersion problem will be studied in greater detail here.
Solutions to the problems of  dispersion in porous media 
have been obtained by many investigators, (Hunt, 1973; 
Hunt, 1978; Mariño, 1978; Basak and Murty, 1977; 
Basak and Murty, 1978; Singh and Murty, 1980; Fattah 
and Hoopes, 1985; List et al., 1990; Hondzo et al., 1991). 
Common to these is the solution of  the dispersion 
equation of  the typical unconfined aquifer of  the Sidi 
Kirayr locality. (Somaida, 1993), has applied the finite 
difference method for a preliminary solution of  the 
dispersion equation for the unconfined aquifer in the same 
locality.  The problem has been modeled and investigated. 
These models, (Somaida, 1993) are tested, and the results 
of  salinity depth profiles are deficient in the qualitative 
and quantitative interpretation of  the salinity profiles. 
In previous research papers, the numerical groundwater 
modeling approach has discussed the significance of  
groundwater as a significant source of  freshwater. 
It is used to quantify spatial and temporal trends in 
groundwater flow and availability as well as to examine 



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the overall effects of  human activity, (Abdalla, 2016; 
Al-Dabbas, 2020; Imagawa et al., 2012; Matić and Srzić, 
2021).
The present study will concentrate on promising 
models, where their results will be compared with the 
real data encountered in the area. The present study 
will be conducted on two concepts. First, the two-
dimensional dispersion equation is derived as applied to 
a typical unconfined coastal aquifer. Second, numerical 
computation of  the studied aquifer’s dispersion equation 
to reach the vertical groundwater zonation based on 
aquifer vertical water quality measurements. The water 
quality results as indicated by electrical conductivity 
measurements are assured by comparison with the vertical 
geochemistry zonation performed at the same aquifer.
Mathematical and numerical modeling can help in solving 
many engineering problems and that can save time and 
effort (Amin and ElZahar, 2022, Sharaan et al., 2022; 
Somaida et al. 2011, ElZahar et al., 2001; Fujisaki and 

ElZahar, 2002).

MATERIALS AND METHODS
Finite Difference Formulation
The general differential equation for the distribution 
of  dissolved salts in groundwater under unsteady state 
conditions in a porous medium is, (Anon, 1965)

Where c = concentration of  the salt in groundwater, Dx, 
Dy, and Dz = the dispersion coefficients, Vx, Vy, and Vz 
= the average pore velocities in the directions (x, y, z), and 
t = time. The first term on the left-hand side of  (Eq. 1) 
depicts the diffusional transport and the second group, 
the convectional transport. 
Considering a two-dimensional problem, (Eq. 1) can be 

Figure 1: Definition sketch for the studied coastal unconfined aquifer

Figure 2: Rectangular mesh for the application of  the finite difference technique



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Am. J. Environ. Clim. 2(1) 1-10, 2022

simplified to:

Equation (2) is a second-order differential equation that 
can be rewritten in the following form:

Where,

It is noted here that the changes in Dx and Dy are 
negligible. In this problem, the concentrations can be 
known along any boundary, hence, it will be a boundary 
value problem. 
A classical method for numerically approximating a 
solution for a boundary value problem for this type of  
equation consists of  replacing each derivative in the 
differential equation with a difference quotient that 
approximates the derivative, (McCormik and Salvadori, 
1968; Wagle and Saff  1986), then:

Where Δx = the spacing in the x-direction and Δy = the 

Figure 3: Volume of  trash cans

spacing in the y-direction, (Figure 1,) and Δt = the time 
step.

However, if  it is assumed that the dispersion coefficient, 
D, does not change with time or distance in porous 
medium and taking equal spacings in the directions X and 
Y (Δx = Δy = Δ), then (Eq. 9) is simplified to:

(Eq. 10), can be used to determine the concentration C 
at any pivotal point in the flow domain and is derived 
according to the following assumptions.
1) One phase flow, the aqueous phase consists of  two 
components water and salt. 2) Fluid is incompressible. 3) 
Homogeneous, isotropic porous medium with constant 
dispersion coefficient. 4) Two-dimensionalnal unsteady 
flow in the porous medium . 

Determination of  Parameters
This can be done through the application of  the finite 
difference method on the actual condition of  the coastal 

unconfined aquifer of  the Sidi Kirayr locality west of  
Alexandria, (Abdel-Mogeeth and Elshazly, 1980), Figure 3.
In this aquifer the water table is determined according to 
measurements of  groundwater levels in the native wells 
scattered in the area and the fresh-saltwater interface is 



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determined theoretically by Ghyben-Herzberg principle 
and practically by meaning the electrical conductivity 
in certain test wells in the area (Abdel-Mogeeth and 
Elshazly, 1980). A typical vertical cross-section of  the 
studied aquifer is shown in Figure 4. The studied  aquifer 
zone is selected along the center line of  the aquifer 
(location of  the hypothetical well, Figure 3) through the 

dispersion zone formed between fresh and saltwater. The 
salinity profile along this line, Figure 5, shows an abrupt 
change of  salinity from 2 mmhos to about 39 mmhos at 
depths from 24-28 m.
However, the dispersion zone in the lower 10 meters 
(28 to 38 m), is considered to be detected and analyzed, 
where the salinity changes gradually from 39 to more than 

Figure 4:  Extent of  fresh and saltwater in the Sidi Kirayr coastal unconfined aquifer, northwestern coastal zone, 
Egypt (Somaida, 1993)

50 mmhos. The selected domain is a rectangular mesh, 10 
m in length and 4 m in width. 
Concentrations are given along the left boundaries, while 
those at the right one are to be computed.
The average hydraulic conductivity of  the aquifer K = 
10 m/day, the effective porosity n = 0.2, and the average 
upward pore velocity Vy = 0.05 m/day, determined 
from the study of  water table levels in the area and the 
use of  Ghyben-Herzberg principle, (Somaida, 2014; 
Wirojanagud and Charbeneau, 1985).
The average pore velocity in the X-direction Vx is 
determined at any point in the flow domain using the 
following equation (List et al., 1990):

where ds  and df  are specific gravities of  the sea and 
freshwater (l.025, 1.0), and X and Y are the coordinates 
of  the pivotal point from the origin which is taken as the 
location of  the hypothetical well, Figure 3.
The dispersion coefficient can be calculated using the 
following equations and the field data as follows:
For transition zone length, (Schmork and Mercado, 1969):
Lmix= Z(c=0.159)-Z(c=0.841) (12)

Where,
Zc=0.159 = depth relative to 0.159 of  the initial 
concentration and Zc=0.841 will be relative to 0.841.
From Figure 5, it is found 11.6 m.
The mixing length as a function of  Zrise (interface rise) 
and dispersivity a1, is given by:
Lmix=2 √(2 a1 Zrise ) (13)

The decline of  the water table in the actual well S2b, 
Figure 3, is 0.125, then according to Herzberg principle, 
Zrise=40*0.12=4.8 m. However, the application of  Eq. 13 
gives 11.6=2 √(2 a1*4.8) , then dispersivity of  the aquifer 
= 3.5 m.
According to Ref  (18), the dispersion coefficient 
varies approximately directly with Vx. The ratio DL/
Vx, theoretically is a constant, called the dispersivity 
having the dimension of  length and in the present study 
= 3.5=DL/Vx . Here Vx is calculated as equal to KI, 
where K = hydraulic conductivity of  the aquifer = 10 
m/day and I is the hydraulic gradient in the transverse 
direction at nearly the middle of  the asphalt road 
(location of  htheypothetical well) = 0.125/46.7=0.0027. 



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Figure 5:  Observed water salinity with depth at the center of  the studied aquifer (Somaida, 1993)

Figure 6. a:  Concentration results and their graphical representation along the studied profile (Model M1)

Figure 6. b:  Concentration results and their graphical representation along the studied profile (Model M2)



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Figure 6. c:  Concentration results and their graphical representation along the studied profile (Model M3)

Figure 6. c:  Concentration results and their graphical representation along the studied profile (Model M3)

However, Vx=10*0.0027=0.027 m/day, from which, 
DL=3.5*0.027=0.095 m2/day.
Numerical Study
A computer program for the solution of  the dispersion 
expression, Eq. 10, is developed. The program required 
the knowledge of  the concentrations at the left side of  
the proposed domain together with distance step ∆, time 
increment ∆t and the diffusion coefficients D. In this 
view, six mathematical models are employed (M1 to M6). 
For M1, M2 and M3,  the increment, ∆ is taken 2 m and 
dispersion coefficients of  0.095, 0.0475, and 0.19 m2/day, 
respectively. The same choice is decided for M4, M5, and 
M6.

• Model M1, Figure 6-a, shows satisfactory 
concentration values with depths which are close to the 

values measured in the field until ∆t=20 days. At time t = 
40 days, concentrations are nearer to the real values in the 
upper and lower 4 meters in the studied aquifer. While at 
t = 60 days, the concentrations are highly oscillating and 
far from the real values.

• Model M2, Figure 6-b, shows satisfactory trends 
but the concentrations are less than the expected values. 

• In model M3, Figure 6-c, the concentrations are 
mostly far from the actual values. High oscillations are 
very evident along the salinity profile.

• Models M4, M5, and M6, Figs. 6-(d-f), indicated 
bad plots which are completely out of  phase, indicating 
the lack of  a general trend for salinity.
However, it is discovered from the studied models that 
the best values of  the aquifer’s dispersion coefficient D 



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Figure 6. d:  Concentration results and their graphical representation along the studied profile (Model M4)

Figure 6. e:  Concentration results and their graphical representation along the studied profile (Model M5)

Figure 6. f:  Concentration results and their graphical representation along the studied profile (Model M6)



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are 0.095 m2/day. The best and most trustworthy results 
for further research in this paper come from model M1.

RESULTS AND DISCUSSION
Hydrodynamic dispersion in the coastal aquifer
It appears that the longitudinal dispersion of  fresh and 
saltwater is active upward in coastal unconfined aquifers 
as explained by (Anon, 1965). Since in coastal aquifers, 
fresh water will be contaminated from saline water 
and the concentration of  fresh water will be increased 
with increasing distance of  flow upward because of  

hydrodynamic dispersion, longitudinal dispersion occurs 
in the direction of  flow and is caused by different 
macroscopic velocities as some parts of  the salt invading 
water move through wider and less tortuous pores than 
other parts, (Bouwer, 1985). The different rates of  
advance produce the typical sigmoid breakthrough curves 
of  the invading saline water at the same distance from 
the front of  the invading fluid. This phenomenon is best 
developed in the coastal aquifer of  the Sidi Kirayr area 
and is observed in the detecting the test well S2b, in the 
Sidi Kirayr locality, Figure 3, which is drilled to a depth 

Table 1: Distribution of  households surveyed by neighborhood, type, number, and size
Depth (m) 36 32 28 24 20 18 16 12 8
Detected EC mmhos, S2b 54.9 48.1 42.6 17.15 12.86 2.75 2.46 2.3 2.2
Computed EC mmhos, M1 50.1 44.5 41.2 19
Computed EC mmhos, M2 50.1 44 41 39

of  38 m, the depth of  water table = 10 m. The well is 
completely penetrating the aquifer and its discharge is 
about 60 m3/day (6.94 * 10-4 m3/sec), Figure 5.
Computed concentration-depth profiles for the studied 
aquifer
Table 1 depicts the profiles from which the concentration-
depth profiles are constructed, as well as the models M1 
and M2, Figure 5, and the real concentration-depth profile 
associated with the actual well S2b, Figure 7. Investigation 
of  this figure leads to the following conclusions:

• Concentration (salinity) increases with depth.
• Both profiles, the real and M1, have the same trend 

in which the relatively sigmoid breakthrough curves are 
observed.

• The profile M2 shows oscillating salinity values 
relative to M1, far from the expected values, which can 
be safely disregarded.
The M1 profile shows satisfactory results which are nearer 

to the actual field values. This is attributed to the proper 
choice of  model M1 and its necessary data as; depth 
increment ∆, time step ∆t and the diffusion coefficient D 
relative to the actual field data from the Sidi Kirayr locality.

Comparison between computed and detected salinity 
depth profiles
For this purpose, Table 1, showing the variations of  
concentrations with depth and salinity profiles in Figure 
7, is discussed and the following conclusions are reached:

• The salinity profiles show the same trend in the 
interval from 24-28 m and are nearly parallel profiles. 
This may indicate the correction of  the approach of  the 
derived dispersion Eq. 10, its solution, and assumptions.

• There is a marked decrease in salinities in the actual 
test well S2b below the computed values. This may be 
attributed to the nature of  the geochemistry of  the water 
in the actual test well S2b.

Figure 7. Correlation of  salinity profiles; Real (S2b), computed M1, and M2



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• In both cases, the rate of  salinity change increases 
with depth, where the actual sea water is found at depth 
of  38 m, (Anon, 1965).
Vertical Zonation of  Groundwater
In coastal aquifers, freshwater is physically defined as 
water having specific gravity of  1.0, which is floating over 

CONCLUSIONS
In the present study, the finite difference method has 
been applied to develop and solve the dispersion equation 
for a typical unconfined aquifer as the case implies in the 
Sidi Kirayr locality, west of  Alexandria, Egypt, where the 
problem has been modeled and investigated. 
Numerical calculations of  the concentration of  water 
with depth in an actual tested well in the area have been 
reached using a well-developed computer program, 
which solves the dispersion equation and obtains the 
water salinity at a depth varying from 18 to 38 m along 
a computed vertical profile at the center of  the lens. In 
addition, electrical conductivity measurements were 
performed in the test well S2b, which is very close to the 
lens’ center. 
When the computed salinity profile below a good example 
was compared to the measured salinity profile in well S2b, 
there was an order of  magnitude agreement. The sigmoid 
breakthrough curves of  both situations exhibited nearly 
parallel trends, indicating that the electrical conductivity 
measurements were successful. Also, this shows that the 
vertical zonation of  groundwater in the Sidi Kirayr locality 
can be distinguished into three characteristic water zones: 
a top fresher water zone (2-4 mmhos), a brackish water 
zone, then a salt water to seawater zone. Finally, below 
this, there is the actual seawater of  the Mediterranean 
Sea above 50 mmhos salinity. The latter results are found 
to be in good agreement with those reached by (Abdel-
Mogeeth and Elshazly, 1980).

Nomenclature
C = Concentration, 
Δt = Time increment,
Vx = Velocity of  flow in the X-direction,
Vy = Velocity of  flow in the Y-direction,
Δx = Distance increment,
D = Dispersion coefficient of  aquifer,
ds = Specific gravity of  saltwater,
df  = Specific gravity of  freshwater,
K = Hydraulic conductivity of  the aquifer,
n = Porosity of  the aquifer,
xi,j = Horizontal coordinates of  the pivotal point,
yi,j = Vertical coordinates of  the pivotal point,
Lmix = Length of  transition zone (mixing length),
a1 = Dispersivity,
Zrise = Rise of  interface measured from reference level,
DL = Longitudinal dispersion coefficient.

REFERENCES
Abdalla, F. (2015). Ionic ratios as tracers to assess 

seawater intrusion and to identify salinity sources in 
Jazan coastal aquifer, Saudi Arabia. Arabian Journal 
of  Geosciences, 9(1). https://doi.org/10.1007/s12517-
015-2065-3.

Abdel-Mogeeth, S. M., and Elshazly, M.M. (1980). 
Characteristics of  the freshwater lens in Sidi Kirayr 
locality, North-western coastal zone, Egypt. Desert 
Institute Bulletin, A.R.E., 30(1), 1-15.

Freshwater 2 Mmhos
Passably freshwater 2-4 Mmhos
Brackish water 4-9 Mmhos
Saltwater 9-50 Mmhos
Extremely saltwater >50 Mmhos

saltwater of  specific gravity of  about 1.025. In terms of  
total salinity, expressed as electrical conductivity/cm at 25 
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classification:

• A zone of  passably freshwater class<4 mmhos or 
(2-4) of  depth 8-20.6 m, in well S2b. The corresponding 
change in salinity is 0.16 mmhos/m.

• A zone of  brackish water (4-9 mmhos), from 20.6 
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• A zone of  salt water (9-50 mmhos) from (22-36 m) 
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• It is noted that in Figure 7, the transition zone of  
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i.e., lies in the saltwater zone.
Although the changes in salinity are 2.93 and 3.1 
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Table 2: Classification of  groundwater according to 
salinity profiles in the Sidi Kirayr are, (Hem 1985)
Freshwater 2 Mmhos
Passably freshwater 2-4 Mmhos
Brackish water 4-9 Mmhos
Saltwater 9-50 Mmhos
Extremely saltwater >50 Mmhos

is recommended that reliance should be made on the real 
data obtained from measuring the electrical conductivity 
in real test well S2b, Figure 3.



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