In modern science and engineering, differential equations are very important. Nearly all known physics and chemistry laws are indeed differential equations. Engineers, in order to investigate systems behavior, it is virtually necessary that they are able to model and solve physical problems with mathematical equations.

1. Soran University
Faculty of Engineering
Department of Petroleum Engineering
Engineering Analysis I Report
Supervised by: Mahfouz Rostamzadeh
Applications of Differential Equations in Petroleum Engineering
by:
Raboon Redar
Fall 2020

2. 1. Introduction
In modern science and engineering, differential equations are very important. Nearly all known physics and
chemistry laws are indeed differential equations. Engineers, in order to investigate systems behavior, it is
virtually necessary that they are able to model and solve physical problems with mathematical equations. In
order to investigate the behavior of the structures involved, engineers need to be able to mathematical equations
and then solve their physical problems. In different areas of engineering and research, differential equations
have wide applications. Modeling to adjust physical quantities, including temperature, stress, displacement,
velocity, gravity, strain and current, the voltage, or the concentration of the gas, as the time or the position
changes, or leads to differential equations.
2. Well Testing
A "well test" is simply a time during which the production of the reservoir is measured, either on the well 's
head or in the surface production facility. The main differential equation is accomplished by combining the
equation of continuity, fluid flow and the state equation. The most complete mathematics has come from
solutions originally designed for the heat transfer analysis for this equation. Fluid flow from pores is like heat
flow from solids directly. Best test solutions usually require radial flow and consistent isotropic formations.
2.1 Continuity
Continuity equation is one of the important equations in oil engineering. The equation it provides the principle
of conservation of mass in the radial coordinates. It explains that the rate at which the mass leaves the system is
equal to the rate at which the mass comes into a system at any constant state phase.
The Continuity Equation: [ ]
2.2 Fluid Flow
The prevailing factor relating to condensation in the pores of the reservoir as the pressure drops involves a
software program to combine the transient radial fluid transient differential equal. Some assumptions had to be
made in preparation of the software implementation as regards the motion of the liquid phase and changes in
radial distribution and recovery. Due to the significant dependence on the mobility of the fluid reservoir and the
related pressure gradients on liquid saturation, this presumption was beneficial to be eliminated and the integral
output reliability improved.
Horizontal Multi-Phase Fluid Flow: ( )
Same thing for the Water and Gas Flow: ( )
( )

3. 2.3 The State equation
In physics and thermodynamics a status equation is a thermodynamic equation which relates the state variables,
which describe the status of matter under certain physical conditions, such as stress, volume, temperature (PVT)
or internal energy.
The general Equation of State for Ideal Gases:
2.4 Diffusivity
Diffusivity equation is one of the major oil engineering equations and for reservoir simulations. The equation is
also used in the testing of well-testing data where time is usually recorded in hours. It is a generally the measure
of the capability of a substance or energy to be diffused or to allow something to pass by diffusion.
Diffusivity Equation:
3. Seismic Surveys
Noise and the useful enhancement of the seismic signal with filters is one of the essential tasks in seismic data
processing. Conventional methods for filtering may be used either by convolution or various mathematical
adjustments. We describe a method for filtering based in comparison to different filtering techniques on the
resolution of differential equations in space and time.
The wave equation:
4. Compressibility
For several, it is important to be aware of the instability of fluid compressibility and pressure and temperature
Calculations of reservoir-engineering. Compression is low and usually considered to be constant for the liquid
phase. Compressibility is neither low nor constant during the gas phase. The factor of natural gas compression
is a measure of gas volume, which is different from the ideal gas behavior. It is an important tool in the gas
industry for direct or indirect measurement of the characteristics of reservoir fluids. Implementation: gas
calculation, gas flow in tubes, measurements for the fluid balance, well analysis, well-simulations, etc. Relation
is made to the volume change per unit volume compressibility for a change in unit pressure.
Oil Compressibility Equation: ( )
5. Permeability
The law of Darcy relates to the concept of the movements of oil, gas and water from oil reservoir which is also
similar to the fluid flow. The fluid movement within the rock is regulated by the rock's permeability. The
permeability of a medium is correlated with porosity but also with pores in the medium and their connectivity
level.
Darcy’s General Law:

4. Conclusion
Different equations in engineering are important because of their usage in various fields of engineering and the
various forms of differential equations. One of them is the petroleum engineering, which uses all these
applications to solve different equations that are used in the solution of problems in engineering.
References
2007. Petroleum Engineering Handbook, Volume 5. [Place of publication not identified]: Society Of Petroleum
Engineers.
Abou-Kassem, J., Farouq-Ali, S. and Islam, M., 2013. Petroleum Reservoir Simulations. Burlington: Elsevier
Science.
Ahmed, T., 2010. Reservoir Engineering Handbook. Amsterdam: Gulf Professional Pub.
Aziz, K. and Settari, A., 2002. Petroleum Reservoir Simulation. Calgary: K. Aziz & A. Settari.