Application of a new Mean Velocity equation of Flow in a natural
stream of 8.5 m3/s, Its Verification and extrapolation: Copper Creck
river, Virginia (USA).
ALFREDO JOSE CONSTAIN ARAGON
Fluvia Tech.
Bogotá, Calle 132A # 19-64 (301-2)
COLOMBIA
Abstract: - The application of a new equation for average flow velocity in turbulence has already been done in a
calibrated channel with precise results [1], so it is interesting to apply it now to a larger natural channel, of 8.9
m3/s, with tracer measurements that They were made in the USA in 1959 and were very well documented
[2][3][4]. This condition of abundance of information, the same as in the Caltech Canal, allows the reliability of
the new equations to be verified in detail, and to achieve substantial improvements in river hydrometry techniques
Key-Words: - Hydraulics; Fick, Chezy-Manning; Models; turbulence, state functions.
1. Introduction
Environmental and Hydraulic Impact studies
currently depend critically on a certain set of data
from water bodies that generally depend, in their
quantity and quality, on the measurement methods
used. Thus, although there are very advanced and
powerful computer programs, and there is refinement
in many equipment and techniques, a great drawback
remains in that there is no direct link between
Hydraulics, geomorphology and dispersive transport,
only at the level of semi-empirical equations, that do
not ensure a necessary generality.
The author has presented a definition of a State
function, involving the mean velocity of flow, as
follows. [5][6]
(1)
Here the “dispersion velocity”, ±vd, acts in both
longitudinal directions from the center of the
Gaussian distribution, and whose module is:
(2)
With, Δ equal to a characteristic Brownian
displacement, and τ equal to the characteristic time of
that displacement, with D as Longitudinal Dispersion
Coefficient.
The definition of the characteristic time, τ,
corresponds to the Brownian mechanism of self-
similarity at all scales established by M. Feigenbaum,
with its fractal constant, δ. [7]
(3)
In the formulation of the mean velocity of flow, U,
the interactions that are considered are the Coulomb
forces of the Van der Waals type.[8].
(4)
Of course, the average velocity corresponds to that
defined in mechanical terms, according to Chezy-
Manning, considering mechanical, Newtonian,
forces [9].
(5)
Where R is the hydraulic radius, n is the Manning
roughness, and S is the energy slope of the flow.
An important characteristic of the new equation is
that it depends on a thermodynamic potential, such
that:
(6)
Received: April 6, 2024. Revised: August 21, 2024. Accepted: September 25, 2024. Published: October 17, 2024.
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This is important for several reasons, since this type
of function is sensitive to the thermodynamic
conditions of the process, and also depends on few
variables, which allows defining a statistical coupling
of the tracer with the flow, since basically the
evolution of Φ, reflects the phase transition of the
solute from solid to gaseous, when the tracer plume
is so diluted (osmotically) that it is almost virtually a
set of water particles, with the particularity that it can
be traced with instruments. [10]
Historically an important link, between Hydraulics,
geomorphology and dispersive transport, was the
proposal by W. Elder [11], who early related the
slope, S, and the depth of flow, h, with the
longitudinal dispersion coefficient, D.
(7)
This equation did not gain sufficient audience, since
at the time it was proposed, it was thought that since
it was based on the vertical distribution of flow
velocity, it did not also represent the lateral
distributions.
In this way, current techniques for studying river
advection and dispersion have lacked tools that
reestablish this link. This is the aim of presenting this
new flow velocity equation.
2. Problem Formulation.
2.1 The evolution of State function: How to
define and apply it.
Based on the above concepts, the state function
can be approximately defined as: [12].
(8)
Here, M is the mass of tracer injected, tp is the peak
time of tracer curve, and γ is a characteristic
parameter of each solute used, and which must be
calculated experimentally:
(9)
Then, it can be established that Φ(t) evolves as shown
in Figure 1.
Figure 1.- Evolution of State function Φ(t).
The steep segment (up to Φ≈ 2.16) corresponds to the
“solvation” process of the solute. The smooth
decaying segment corresponds to the evolution of the
ions, which diffuse until at Φ≈ 0.38, almost all of
them are in the gas phase, and Φ(t) is considered to
describe the turbulent evolution of the flow.[13]
Figure 2.
Figure 2.- Turbulence described locally by a State
function Φ(t).
The blue “patches” seen in this figure correspond to
linearity fluctuations related with discontinuities of
the homogeneity on the “Coarse grained”
representation, that the “dynamic equilibrium”,
ideally would impose on the Probability Distribution
of the channel.[14][15]
This effect arises from the fact that water flows do
not ideally comply with the “linearity” of non-
equilibrium thermodynamics, deviating somewhat
from “statistical sufficiency”. [16][17][18] This fact
means that the variances of the magnitudes in the
fluvial processes are small but not zero (Principle of
minimum entropy production).
2.2 Efficient planning of Advection-Dispersion
measurements using tracers.
These considerations allow planning measurements
in turbulent channels of a certain large size, in which
other methodologies fail due to not having precise
extrapolation mechanics, such as being able to know
where the "Complete Mixing" condition occurs, as
the transition is called. to “ideal gas” of the tracer (Φ≈
0.38) where the area in which the channel already
meets its “assimilation” condition is conventionally
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delimited, and where the location of physical,
chemical and biological monitoring equipment is
located by the eventual homogeneity of their
readings.
This feature of the State Function is very useful as it
allows practical calculations of turbulent flow, at
convenient distances, and using minimum amounts
of tracer, since it is not required to “saturate” the flow
(as in conventional methods). All this is possible
without entering into conventional discussions about
the non-linear nature of the process [19]. Figure 3.
Figure 3.- Extrapolation by means of State
function Φ(t).
With these criteria and techniques, it is feasible to
plan an efficient measurement campaign in which a
series of Advection, Dispersion and geomorphology
data can be provided, which guarantee very precise
modeling procedures.
2.3 Modified Fick equation: realistic
representation of asymmetric tracer curves.
The classical theory of one-dimensional transport for
conservative solutes is expressed in the Fick
equation, with M the mass suddenly injected into the
flow, Ayz the cross-sectional area of the tracer flow,
U the average velocity and D the dispersion
coefficient.
(10)
This definition considers that D does not depend on
time (constant value), so the shape of the theoretical
curve does not represent the experimental bias that
appears.
To restore the faithful representation of reality, the
Coefficient D must be cleared from equation (4), and
replaced in equation (10), where τ≈0.214*tp, using
Feigenbaum's development [20][21]:
(11)
In this expression Φ(t) fulfills the function of “form
factor” while the degree of asymmetry is determined
by it, and the irreversible losses are represented by
U2. This means that the Longitudinal Dispersion
Coefficient is a non-linear function of time since the
factors that define it are in turn a function of time.
From this we obtain “modified” Fick equation, with
Q as the flow rate:
(12)
When the mass of the tracer is in milligrams or
micrograms per liter, the flow rate is in Liters per
second.
2.3 Flow velocity and tracer centroid velocity.
The question usually arises about what velocity data
to use when describing the movement of the flow: If
it is related to the peak of the distribution, tp, or if it
is related to the time of centroid of it, ts, which is
greater than tp. Figure 4.
Figure 4.- Center and centroid of the distribution.
When the solute is injected into the flow, the dipolar
water molecules attract the tracer particles,
destroying its crystalline structure (solvation or ion-
dipole interaction). From there, you will have a set of
ions that increasingly separate from each other more
and more, which leads to a decrease in concentration
as a function of time. This concentration produces a
braking effect on the tracer ions themselves, which
can be modeled as a mechanism of ion-ion
interaction.[22]
For this reason, you must distinguish the velocity of
the “mass centroid” of the tracer plume from the
velocity of the flow itself.
(13)
And
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(14)
As the braking effect of the ions on each other
depends on the concentration, and this decreases with
time, the two velocities tend towards a common
value. Figure 5.
Figure 5.- Behavior of v(ts) and v(tp)
For the “Complete Mix” condition, the difference
between them is approximately 13%.
2.4 The kinematics of the tracer plume: Source of
symmetry for a Lagrangian observer and of
asymmetry for a Eulerian observer.
For a moving (Lagrangian) observer who goes over
the peak of the distribution, with velocity U, as there
is no displacement of the mass centroid, consequent
to the fact that there is no composition of velocities
between U and ±vd, where vd is the modulus of the
velocities of dispersion, side by side on the X axis.
Figure 6.
Figure 6.- Symmetric tracer curve measured by an
Lagrangian observer.
For this observer, the distribution equation that is
valid is one that does not consider the kinematic
composition analyzed, i.e. the distribution has not
bias (Φ function is not present) but does consider the
effect of irreversible losses.
(15)
For an observer fixed on the shore (Eulerian), who
watches the passage of the plume, this does make the
composition of velocities U and ±vd, in such a way
that the front of the plume advances with greater
speed than the rear part, which It goes at a lower
speed, hence the asymmetry of the curve. Figure 7.
Figure 7.- Asymmetric tracer curve measured by an
Eulerian observer.
In this case, the modified equation of Fick (12) is
valid, showing the bias.
2.5 Extrapolation of measurements through the
State Function and the new equations.
Although software systems have made great progress
and efficiently support modeling in engineering and
environmental studies, a serious problem remains in
this field, which is the absence of efficient
measurement methods to provide timely,
homogeneous and precise data. In particular the
methods of obtaining advection and dispersion data
are archaic to say the least. Modern current meters
and Doppler profilers, although digital, are
essentially based on the principle of conservation of
momentum, and provide “local” data, without
connection to the flow as such.
Tracers, on the other hand, are based on the principle
of energy conservation, and can provide “general”
flow data, given their thermodynamic foundation.
However, the current methods and equipment of this
technique do not have theoretical tools that expand
their field of action, and the technologies are coupled
to this way of understanding measurement processes.
Function the evolution of the plume as such, there is
a basic uncertainty that is attenuated with the
application of multiple equations and empirical
values, which, without detracting from their value,
reflect conditions. “local” that are often not
applicable.[23][24]
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On the contrary, the methodology based on the State
Function and the average flow velocity equation
based on Coulomb interactions, allow efficient
planning of monitoring campaigns, promoting the
obtaining of a series of coherent and precise data for
feed the models.
2.6 Extrapolation procedures.
A.- Advection, Dispersion, geometric and
geomorphology parameters interconnected.
A main objective in the detailed study of a channel is
to have the State Function, which allows the
calculation of parameters such as the Solute
Concentration at a given distance, and others
dispersive and advective parameters as will be
analyzed below.
Apart from having the equation of the State Function
itself, equation (7), a first step is to be able to use it
to make an interpretation that relates it to the basic
Geometry and geomorphology data. To do this, the
following operations are carried out on the identity of
the two one-dimensional mean velocity equations (4)
and (5):
(16)
To make a coherent development, “n”, the Manning
Roughness, must be replaced by its dimensional
equivalent dependent on the hydraulic radius and
time, with “k” a constant that allows solving
numerical proportionality. [25]
(17)
Therefore, we can solve for Φ:
(18)
As you can see, this equation is equivalent to
equation (8) that defines the evolution of the state
function. From here it can be established that:
(19)
From this it follows that knowing all the data of the
tracer (dispersion and advection factors), one can
solve for a hydraulic and geomorphological variable
based on the other.
Normally the easiest thing is to establish the
Hydraulic Radius (by bathymetry), and clear the
slope, which is more difficult to establish.
B.- Calculating the area of cross section of flow by
means of C(X) distribution.
Apart from this evident improvement in the ability to
interconnect parameters that are measured by the
tracer curve and “external” parameters such as slope
or hydraulic radius, which are measured or estimated
by other means, the state function and its related
equations, allow determining other parameters such
as the area of the cross section through the area under
the curve of tracer concentration as a function of
distance, which will also have bias since it is another
version (spatial and not temporal) of the curve seen
by the Eulerian observer on the shore. Figure 8.
Figure 8.- Calculating Area of cross section by using
C(X) distribution.
La distribución requerida, C(X), se escribe como:
(20)
Y el area de la sección transversal es expresa como:
(21)
3. Application of new method: A
case of study.
3.1 The Copper Creck stream experiment in
USA, in 1959. [26]
between 1959 and 1961. This channel in the studied
section is not small to the extent that it has 8.5 m3/s
and an average width of 18.0 m, and an average depth
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of 0.84 m. Curves of the 1-60 experiment, with
radioactive isotope as tracer is shown as in Figure 9.
Figure 9.- tracer curves at 4 and 6 sites.
A radioactive isotope was used as a tracer, gold
trichloride -198 (AuCl3) which has the following
advantages over other isotopes: A: allowing a
maximum manageable concentration, B. - shorter
half-life time (decay rate) which reduces the risk of
contamination. C.- Reasonable cost.
Radioactive tracers are chemical species that emit
particles in a way that are detected by “Geiger
counters”, and are measured, in each species by: A.-
The decay constant, λ, specific for each species,
which measures the % of atoms that disintegrate in
the unit of time, B.- Half-life, T, which is the
reciprocal of λ. C.- Activity, A, is the number of
atoms disintegrated per second, which is measured in
Curios (Cui), referred to a unit of 3.7*1010
disintegrations per second.
Unlike mass, when an “Activity” is injected in flow,
(measured in Cui) the active ingredient already has
the condition “per second”, that is, only
“Cui/volume” (Specific Activity) is required at the
measurement points. [27]
For the experiment (Series) 1.60 at Copper Creck, 6
measurement sections are located, of which sections
4 and 6 are chosen for measurements with
instrumentation, whose photographs from the time
are shown. Figure 10.
Figure 10.- Probe and data logger for signals.
Using the Second tracer curve in Figure 9, the
“Activity” injected at the beginning of the
experiment is initially calculated, through graphical
integration, which is data that does not appear in the
documentation, and which is important for the
calculations. Figure 11.
Figure 11.- Graphic integration for injected Activity.
(20)
Series 1-60 was documented in great detail regarding
its geometric and geomorphological characteristics.
In English system. Table 1.
Table 1.- Geometric, hydraulic and geomorphological
information in English System.[28]
This data is converted into I.S in Table 2.
Table 2.- Geometric, hydraulic and geomorphological
information in I.S. System.
Parameter
Section 4
Section 6
Distance from
injection
X1≈2400.0 m
X2≈4100.0 m
Discharge
Q1≈ 7.62 m3/s
Q2≈ 8.72 m3/5
Mean Velocity
U1≈ 0.53 m/s
U2≈ 0.45 m/s
Area of Cross
section
Ayz1≈14.4 m2
Ayz2≈19.3 m2
Hydraulic
Radius
R1≈0.80 m
R2≈0.89 m
Mean depth
H1≈0.814 m
H2≈ 0.902 m
Mean width
W1≈ 17.7 m
W2≈ 21.4 m
Slope
S1≈0.00137
S2≈0.00130
The tracer injection characteristics are shown in
Table 3.
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Table 3.- Tracer information in I.S. System.
Parameter
Section 4
Section 6
Injected
Activity
378600 μCi
378600 μCi
Specific
Activity
As≈ 1.13 (μCi/pie3)
≈35.3 (μCi/m3)
As≈ 0.59 (μCi/pie3)
≈ 20.9 (μCi/m3)
Φ (t)
0.257
0.210
tp
4727 s
8596 s
τ
1016 s
1848 s
D
9.42 (m2/s)
8.79 (m2/s)
3.2.- Calculations of some parameters related with
tracer.
A.- Calculation of “γ” for tracer AuCl3-198, using
equation (9), changing Concentration for specific
Activity on experiment in 4 th section:
(22)
B.- Calculation of “γ” on experiment in 6 th section:
(23)
Using the values corresponding to Section 4, the
factor that affects the State Function as time
dependent can be initially found, according to the
modified equation (8):
(24)
Therefore, Φ(t) remains:
(25)
The next step is to check the values for t4 and t6:
(26)
And.
(27)
These calculated values correspond well to Table 3.
Therefore, it is accepted that the State Function for
the analyzed experiment is correct as:
(28)
3.3 Extrapolation of calculations using Φ(tp4) in 6
th section.
A.- Specific Activity in 6 th section.
It is interesting to verify the specific Activity in
Section 6, using equation (9), but using the parameter
γ4:
(29)
This is an acceptable result for hydraulic calculations,
with an error of 9%, which could be improved by
using an average of the “γ” constants.
B.- Verification of mean velocity at 4th and 6th
Sections, using new equation.
(30)
And
(31)
These average velocity data at the two measurement
points are accurate with respect to the experimental
data in Table 2.
3.4 Calculation of the cross-sectional area, Ayz, in
Section 6.
To calculate this area, based on the distribution C(X),
we use the equation (19, calculated in a fixed time
and varying the distance, as shown in Figure 8. It
should be considered that this calculation is simply a
spatial description of what the observer sees. He is
Eulerian at a fixed time, and therefore maintains its
own bias.
Using the equation (19) Its model in EXCEL for
X6≈4130 m is shown in Figure 12.
Figure 12.- Eulerian Distribution C(X).
Using Excel, the Area under the Curve of this curve
is calculated.
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(32)
The area Ayz itself is then calculated with equation
(20).
(33)
This approximate value is of the same order as the
data that appears in Table 2, Ayz≈19.3 m2.
This means that the channel has assimilated 91% of
the tracer (or pollutant) when the transport has been
carried out in a time of 8596 s, at the measurement
point in section 6. Figure 13.
Figure 13.- Assimilation capacity of 91% in section
6th.
This figure can be a practical criterion to measure the
pollution assimilation capacity of a natural channel at
a given site with respect to a remote discharge.
3.5 Calculation of centroid time, “ts”, for tracer
curve.
It is important to know the centroid time, ts, of the
tracer curves and compare them with the peak times,
tp, in order to estimate the velocity of the tracer cloud
and compare it with the flow's own velocity,
according to what is stated in 2.3.
To do this, the tracer curve must be established in
Excel, using equation (34) at first site, where ts and
tp will have their maximum difference. And with the
corresponding dispersion and advection parameters
in tables, , applying to the new Fick equation (12).
Figure 14.
It can be seen that ts4 is almost equal to tp4, being
3% larger. It is evident that there is already a
“statistical coupling” of the plume with the flow.
Figure 14.- New Fick distribution for 4th section.
(34)
It can be seen that ts4 is almost equal to tp4, being
3% larger. It is evident that there is already a
“statistical coupling” of the plume with the flow,
ensuring that the calculated data is correct, describing
the complete flow.
3.6. Calculation of hydraulic data from dispersive
data.
From equation (19) in which the mass, M, must be
changed by the injected activity, A:
(35)
First, find the value of “k”:
(36)
For section 4th (initial) we have:
(37)
Now, with this value, equation (34) is completed and
applied to the second measurement site (Section 6),
to calculate the slope at that subsequent site:
(38)
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This value of slope has the same order of magnitude
as the experimental value.
3.7 Calculations in large rivers.
One strategy may be to adjust a sufficient mass of
tracer, M, and place two fluorimetry measurement
devices, E1 and E2, in sequence, so that several state
function data can be obtained. With equation (8) and
the adjustment of the experimental data, an
extrapolation can be made to the required distance,
D, in the section of the channel under study. within
reasonable limits, which allow the tracer
concentration curve to be modeled, and from there
the corresponding calculations can be derived, as
explained in this article. The variances of the data at
the extrapolation point will be minimized if it is
considered that in principle, the river is in “Dynamic
equilibrium”. Figure 15.
Figure 15.- Extrapolation of measurements by means
of State function.
In the case being studied, applying the approximate
model for Φ(tp), from equation (8), the comparison
is made with the experimental data as shown below.
As you can see, the data is of the same order.
Table 4:
Table 4.- Comparison between Φ(tp) calculated with equation
(8), and the experimental data in Table 3.
Parameter
Section 4
Section 6
Φ (t) experimental
0.257
0.210
Φ (t) model,
equation (8)
0.250
0.205
tp (s), approx..
4730 s
8600 s
4. Conclusions
1.- The key to revealing the values of the hydraulics
in a natural cause is to have a state function such as
Φ(t) that allows calculating the various parameters
involved, both dispersive and advective, and also
making an approximate connection with the
geomorphological parameters.
2.- With this theoretical tool it is possible to
extrapolate calculations to greater distances to which
tracer tests are carried out, which in this way will
require less mass of tracer and facilitate the
experimental part.
3.- A criterion is developed to know when a natural
flow has assimilated (diluted) the mass of
contaminant, based on the calculation of the
percentage of the cross-sectional area covered by the
tracer.
4.-The method presented based on a new equation of
the average velocity, allows connecting and
calculating the most important parameters of river
mechanics, ensuring precise and statistically
homogeneous values to successfully feed the models.
5.- The method is applied to examine in detail an
experiment documented by the USGS in 1963, with
satisfactory results. These results suggest that the
presented method can be applied to solve the
obtaining of critical data for the modeling of large or
medium-sized channels.
References
[1] Fischer H.B. The mechanics of dispersion in
natural streams. Journal of Hydraulics Division.
November, (1967).
[2] Godfrey R.& Frederick B. Dispersion in natural
streams. USGS Open file. (1963).
[3]Fischer H.B. Dispersion predictions in natural
streams. Journal of Sanitary Eng. October, (1968).
[4] Constain A. Verificación del transporte de
trazador radiactivo usando un Coeficiente función del
tiempo en cauces naturales. Dyna, No.175. Medellín,
(2012).
[5] Constaín A., Lemos R. & Carvajal A. Tecnología
IMHE: Nuevos desarrollos de la hidráulica. Revista
Ingeniería Civil, CEDEX, Madrid. No. 129. (2003).
[6] Constain A. Definición y análisis de una función
de evolución de solutos dispersivos en flujos
naturales. Dyna, No. 175. Medellín, (2012).
[7] Stewart I. ¿Juega Dios a los dados? Grijalbo-
Mondadori. Barcelona. (1991).
[8] Karapetiants M. & Drakin S. Estructura de la
sustancia. Editorial Mir, Moscú.(1974).
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[9] Vennard J. Elementos de la mecánica de fluidos.
C.E.C.S.A. México. (1965).
[10] Constaín A. The Svedberg number, 1.54, as the
basis of a State function describing the evolution of
turbulence and dispersion. Chapter Intech Open
book, London. To be published next.(2024).
[11] Elder J:W: The dispersion of market fluid in
turbulent shear flow. Journal of fluid mechanics. 5.
Part 4. May (1959)
[12] Constaín A., Lemos R. & Carvajal A. Ibid
(2003).
[13] Constain A. Ibid. (2024)
[14] Penrose R. Los ciclos del tiempo. Editorial
Debate. Barcelona. (2012)
[15] Leopold L. & Langbein W. The entropy concept
in landscape evolution. USGS Report. (1962).
[16] Constain A. Ibid. (2024).
[17] Annila A. & Makela T. Natural patterns of
energy dispersal. Physics of life review. 7(2010).
[18] Annila A. & Ketto J. The capricious character of
nature. Life. (2012).
[19] Anderson P.W. More is different.
Science,177(4047). (1972).
[20] Constain A. Ibid. (2012)
[21] Constaín A., Peña G. & Peña C. Función de
estado de evolución de trazadores, Ф(U,E,t), aplicada
a una función de potencia que describe las etapas de
la turbulencia. Revista Ingeniería Civil, CEDEX,
Madrid. (2022).
[22] Damaskin B. & Petri A. Fundamentos de la
electroquímica teórica. Editorial Mir. Moscú. (1978).
[23] Constain A. Ibid. (2024)
[24] Constain A. , Peña G. & Peña C. Dispersion and
turbulence : A close relationship unveiled by means
of a State function. Itegam-Jetia. Manaus, V.7. no.
30. (2021).
[25] Chow V.T. Hidraulica de canales abiertos.
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[26]. Fischer H.B. Ibid. (1968).
[27] Stein J. Isotopos radiactivos. Editorial
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[28] Godfrey R.& Frederick B. Ibid. (1963).
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EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2024.4.11
Alfredo Jose Constain Aragon
E-ISSN: 2944-9006
104
Volume 4, 2024
