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\author{GEORGIA MOUDIOTI}
\title{MATHEMATICS AND GEOSCIENCES}
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\section{introduction}
Traditionally, the science of geology has had a relatively minimal mathematical
application in comparison to other sciences such as physics, chemistry and mathematics. This
trend has been because of the importance that was attached to the compositions of the different
rock types, their mineralogy and consequently their relative ages in terms of the geological time
scale.
Basically, geology can be conveniently divided into three categories namely; the
"Hard" Rock geology; the "Soft" Rock geology and the "Applied" geology. The subject
matters that fall within the "Hard" Rock geology category, are Economic geology,
Geochronology, Geochemistry, Petrology/Petrography, Structural geology, Mining geology,
Mineral exploration, etc.
Subjects that fall within the "Soft" Rock geology are: Sedimentology, Sedimentary
geology, Petroleum geology, Paleontology/Pateocology, Economic geology, Organic
geochemistry, Structural geology, Stratigraphy, etc
It would be observed that in the grouping of the subject matters under both categories
there is an unavoidable number of subjects that are common to both groups. On the other hand,
the subjects that can be considered to fall within the category of "Applied" geology are;
Engineering geology, Rock Mechanics, Soil mechanics, Geotechnics, Hydrogeology and
Geophysics. It should be pointed out here that the subjects of Engineering geology, Rock/Soil
mechanics and Geotechnics are very closely related to the disciplines of civil and mining
engineering.
\section{Applications of mathematics to the various branches of geoscienses}
The role or importance of mathematics in each of the three categories of the geosciences
will be looked into in this section of the paper. In doing this;, attention will be paid only to the
relevant subject matters in each of the first two categories while the third category will receive
full treatment since it is entirely made up of subjects strongly dependent on mathematical
applications.
\section{"Hard" Rock Geology}
One of areas in which mathematics is applied in the hard rock geology category is
mining geology. In every mineral exploration programme, it becomes necessary to estimate the
amount of ores (that is, ore reserves) present before the mine is established. This process requires
the knowledge of the lateral extent, the depth of the ore body and the variations of thickness over
the extent of the deposit. Integral calculus is often employed in the determination of the volume of
such ore bodies and hence the computation of such ore reserves. Even when the mine is to be
developed, knowledge of the slope angles of the open-pit is required, and this is usually obtained
through mathematical formulae that incorporate elements of the mine geometry and the material
properties of the mine rock.
In the subject of Geochronology, the ages of rocks are often determined through
radio-metric dating techniques. These techniques, which utilize the half-lives and decay
constants of radioactive materials, employ some mathematics.
The subject of structural geology, which falls within both "Hard" and "Soft" rock domains,
employs quite a bit of basic mathematics. The mathematics is involved in the computation of structural
trends e.g. (fractures and lineation), dip angles, plunges, thickness of rock strata, etc.
\section{"Soft" Rock Geology}
The sediments that constitute the bulk of the "Soft" rocks are deposited through the activities of
waves, wind, rapidly flowing currents, like water, wind among others, and as such much of the
mathematics involved in this category is of the "Dynamics" type.
Sedimentology is a subject in the "Soft" rock category that deals with sedimentary processes.
These processes are directly related to flow velocities of streams and rivers and the rates of stream
sedimentation and scour. The distance from source and other factors of a particular sedimentary deposit
can be inferred mathematically from the sizes and geometries of the grains of sediments within that
particular deposit. This aspect of geology is receiving a lot of attention from computer applications, due to
the size of the data involved in research.
Another area within the "Soft" rock geology category that has quite a bit of mathematical input
is petroleum geology. Though it is of a more definite nature, it tends to be similar in many ways to
petroleum engineering. Here, mathematical techniques abound in the estimation of oil reserves, mode and
rate of oil flow within source rocks in a given oil field.
Even the subject of micropaleontology which has traditionally been that of identification of the
various fossils present in various rock types and age determinants of the various rock based on the relative
ages of the fossils contained in them, has in recent years, no longer escaped the application of basic
mathematics. The degrees to which the various shapes and sizes of fossils have been distorted, are now
used to compute the orientation, the velocity of stream flow and the energy level of pale currents in the
ancient environments in which these fossils and rock assemblages once existed.
\section{"Applied" Geology}
Perhaps the grates needed for mathematical application in geosciences is felt in the "applied"
geology category. In engineering geology, which is basically the application to civil and mining
engineering, the engineering properties of naturally occurring earth materials such as rock aggregates,
gravels, etc. are critically evaluated and applied to constructions in mining and civil practices. In this, case,
the emphasis is placed on the mechanics of earth materials in terms of their compressive, tensile and shears
strengths. Of particular note in this critical evaluation are the forces and moments that act upon these earth
materials during their use in construction and efforts made to ensure that the. superimposed forces do not
exceed their natural strengths.
Soil and Rock mechanics follow very closely set patterns of study in traditional or classical
theories of strength of materials in steel structures, etc, in civil engineering practice. The need to introduce
these courses arose from the fact that many failures were recorded in structures that was adequately
designed in terms of basic structural mechanics in civil engineering practices. However, it was observed
during subsequent post -mortem examination of the failed structures that little or no geological inputs were
inculcated into their initial designs. The relevance of mathematics in these subjects-Rock/Soil mechanics is
quite substantial to say the least.
Another area in the "applied" geosciences where the input of mathematics is felt is
Hydrogeology. In this case, the hydraulics of flow in both surface and underground water is studied with
the aid of basic principles of mathematics. In surface water studies, both laminar and turbulent flow
regimes are treated. In flow through porous media in ground water studies, the flow regime is mostly
laminar while laminar and /or turbulent regimes observed in flow through fractured media.
Geophysics is another area in the geosciences where mathematical applications are very
relevant. In this case, the physics of the earth is treated extensively. In a nutshell it involves the
mathematical treatment of the effects of gravity, magnetism radioactivity and seismcity on deeply located
ores and mineral fuels in the earth's crust.
\section{Conclusion}
From the foregoing discussions, we can now observe that mathematics is becoming more and more
relevant in the various fields of geosciences. The low-key role of mathematics in traditional geosciences such as
"hard" and "soft" rock geology categories, is giving way gradually to a more rigorous mathematical approach
in the relatively new "Applied" geology programme.
It is hoped that before long, almost all aspects of the geosciences will involve more mathematical
input in both research and teaching. This will be more than welcome, bearing in mind the present availability
of micro computers at our own disposal.
\section{References}
\begin{thebibliography}{5}
\bibliographystyle{plain}
\bibitem{Ferguson}
Ferguson, J.\textit{Mathematics in Geology.Unwin HymanHLtd.},UK, 1998
\bibitem{Nibir}
Nibir, M.I. \textit{Fow and strain patterns at the terminations of tapered shear zones.},Structural of Geology, 2002
\bibitem{Oluwatosin}
oluwatosin, G.A. \textit{Variability of surface properties of an ultisol as fuction of topography in sandstone landscape of south western Nigeria},global of Journal of pure and applies scienses, 2007
\end{thebibliography}
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