The Mathematical simulation of the filtration of the fluids in the oil field
Dr. Djavanshir Gadjiev
Education: University of Tennessee, Knoxville
Russian Academy of Sciences (RAS)
Employment: Collier Schools District
Dual Enrollment Edison State College (ESC)
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GEOMETRY WILL DRAW THE SOUL
TOWARD TRUTH AND CREATE THE SPIRIT
OF PHILOSOPHY.
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SCIENCE WITHOUT RELIGION IS LAME,
RELIGION WITHOUT SCIENCE IS BLIND.
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The natural Phenomena or the natural Systems
change over time. By applying principles of
mathematics to such systems to build a
mathematical model, which change in time the
scientist and engineers develop better
understanding of problems in biology,
chemistry, technology, geosciences and
economics and other fields.
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Most PDE equations based on a conservation law.
The physical system evolves with the measurable
parameter(s) of a system. According to a
conservation law the systems such as the
conservation of mass for example, relate to the
point that the mass of undisturbed system of
substances (closed system) remains constant.
Next example is the conservation of energy and
the conservation law states that the total amount of
energy of the isolated system remains constant- by
1-st Law of Thermodynamics.
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Heat as energy that is transferred from one
substance to another, e.g., such as the heat
waves are coming from the Sun.
Vibrating String is an elastic string. A string is
in a balance and can move only in a vertical
plane.
Porous Media Flow- it is a matrix with multiple
pores and throats, which tend to narrow tubes
where fluid can pass through.
Darcy’s Law to model the filtration of the water
in a vertical homogenous sand.
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Laplace ‘s equation: ∆u = ∑∂ₓₓu=0 Solutions to
Laplace’s equations are called harmonic
functions
Heat Equation: ∂ₓₓU - ∆U = f
Wave equation: ∂ᵻᵻU - ∆U = f
Filtration of the fluids(gas/oil/water) in the oil
field: ∂ₓU + ∂ᵧU = m ∂Uᵻ
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1) Oil is important. The oil’s significance is
shocking since Oil/gas power equals to almost
100% of all transportation. The Transportation,
in turn, directly accounted for 1/6th of world
GDP in 1997 and is heavily involved in every
other type of economic activity. Oil is about as
much important to the developed world as
agriculture. It’s truly a condition for the
continued existence of most of humanity today.
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2) The world’s oil & gas transport’s
infrastructure is a globe-spanning consisting of
the spider-web of pipelines and shipping
routes. The natural gas distribution pipelines in
the US alone could stretch from the Earth to
the Moon 7-8 times. There are multiple
thousands of miles of the pipe-lines on the
planet to distribute crude oil, refined products,
and natural gas. Consider this: if your home
has natural gas heat, it is connected via a
continuous network of pipe-lines to tens of
thousands of wells drilled into
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the subterranean rock strata that were laid
down ten of millions years ago.
About 40% of all seaborne cargos are oil , and
there is literally more seaborne cargos at any
given time (by weight) than there are fishes in
the sea. Oil is in transit for a much shorter
amount of time than the lifespan of most fishes,
so the total amount of oil that moves via water
each year is much greater than the total amount
of fish biomass.
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) Unfortunately, at this time it is impossible to
technologically substitute the oil industry.
The oil/gas is critical now, since there are no
viable replacement of fossil fuels in our
lifetime. We hope that the renewable sources
of energy can replace oil within a few of
decades.
However, there is no reason at this time to
think about that any feasible amount of
renewable sources of energy may substitute
fossil fuels in the offing.
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The oil/gas is critical now, since there are no
viable replacements for this type of energy in
our lifetime. We hope that the renewable
sources of energy can replace oil within a few
decades.
The utilization of wind energy and solar
energy are growing in nowadays.
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However, the use of renewables such as wind and solar
energy is a small percentage of the total world energy
consumption. The utilizations of renewable sources of
energy is increased only by 0.07% from 1973 to 2009.
The World oil production was 82 million barrels per day in
2010.
The World wind power production in 2010 was 0.3 watthours . Averaged over a year, that’s about 34 giga-watts.
The World solar power production in 2010 was 0.03 watthours . Averaged over a year, that’s about 3.4 giga-watts.
So, world energy production from oil alone is 2 orders of
magnitude higher than the wind power, and 3 orders of
magnitude higher than the solar power.
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The Newton-Raphson Method is based on the
idea of the approximation of the graph of the
function y =f(x) by the tangent lines:
x²=x¹ -f(x¹)/fₓ(x¹)
x³=x² - f(x²)/fₓ(x²)
Since we assumed that f´(r)≠0 this process will
continue. This process will generate a sequence of
the solutions {xᵗ}, which approaches to the solution
of the system of non-linear equations.
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In the developed mathematical model there were
used the optimization techniques of the Bellman’s
Dynamic programing theory:
We can find the optimal allocation of the
perforated zone in the oil field in order to
maximize the total of the oil extraction and to
extend the economic life of the oil field since the
cone development reduces drastically the
economical Life. The cone development in the oil
field attributes to 30% -40% of the fossil fuels such
as oil and gas, which left in the residual forms in
the pores and it is impossible to further to extract
the fossil fluids at all.
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The Natural Phenomena such as the Earthquakes
or Volcano eruptions or the development of the
tornadoes/hurricanes- all these natural events can
be described by the PDE’s.
The application of the mathematical models
described by the PDE’s led to the computer’s
simulations of these natural disasters.
The computer simulation led to betterment of our
understanding of the physical processes involved .
The computer simulation alongside with the
mathematical modeling gives us a chance to make
prognosis when and where we may expect a
disastrous event to be fully prepared to.
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The teaching of Mathematics in High
Education in 21 century mostly based on the
textbooks which are slightly different from the
textbooks used in the middle of the 20th
century. The textbooks doesn’t give the clear
connection between the abstract mathematical
theory and the real-life situation, e.g. such as
the polynomial functions and equations and
solutions to them are the result of the solutions
to PDE’s. The students must understand how
these equations and functions are wellconnected to the real-life phenomena.
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There are serious issues in Higher Education (College or
University system) there in nowadays: it is a rise of tuition ,
which is not comparable to the quality of the education and
the expectations the students have after obtaining their
degrees for the prospective employment.
There are crisis in Mathematics as the Science there exists,
too: The development of the abstract Theory in Mathematics
is far ahead of the applications of the Theory in real-Life
Situation. Moreover, there is a wide gap in terms on how to
reflect the newest theoretical abstracts in the existed wide
variety of the textbooks. Mainly, the textbooks can be used
as the reference textbooks and the most textbooks does not
challenge a learner. In such case a role of an Instructor is
insurmountable.