THE MATHEMATICAL MODELING OF THE

NATURAL PHENOMENAThe 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)

PLATO

GEOMETRY WILL DRAW THE SOUL TOWARD TRUTH AND CREATE THE SPIRIT OF PHILOSOPHY.

ALBERT EINSTEIN

SCIENCE WITHOUT RELIGION IS LAME, RELIGION WITHOUT SCIENCE IS BLIND.

Language of Math to Describe Natural Phenomena

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.

Mathematical Modeling by using Partial Differential Equations (PDE)

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.

Applications of PDE

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.

Examples of PDE

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

Top Oil Industry Facts

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.

Top Oil Industry facts

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

Top Oil Industry facts

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.

Top Oil Industry Facts

) 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.

Top Oil Facts

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.

Top Oil Facts

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 watt-hours . Averaged over a year, that’s about 34 giga-watts.

The World solar power production in 2010 was 0.03 watt-hours . 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.

PDE which describes the filtration of the gas –oil-water at a singular oil-well.

The allocation of the undisturbed fluids in the oil reservoir

The geometry of the Cone development near the perforated zone of the oil-well.

The development of the gas-water cone at the perforated oil well

Remotely operated subsea system, which enables access to a reservoir of oil up to

several kilometers.

The System of the polynomial equations to which the Integral equations of the original

PDE’s were reduced

Mathematical solution to the system of PDE of the gas-oil-water system

The application of the Mathematical Modeling of the Natural Physical Processes described

by PDE’s in Pre-calculus, College Algebra and Calculus .

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.

The Mathematical Model as the Optimization model to forecast the Economic Life of the Oil

well.

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.

How can we use the mathematical model of the natural systems?

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.

Teaching mathematics at the High and Higher Education

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 well-connected to the real-life phenomena.

Crisis in Mathematics

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.