Drilling or digging for oil has occurred in one way or another for hundreds of years. The Chinese, for instance, invented a bamboo rig to obtain oil and gas for lighting and cooking. Bore hole drilling is divided in to 2 main categories. They are onshore drilling and off shore drilling. Drilling methods have been developed over the time for different purposes and for different environments to increase its efficiency but only in the last 40 years has humankind been able to efficiently extract petroleum even from beneath the ocean as shown below.
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Figure 2: Offshore types and onshore as shown at [4]
Drilling types are categorised depending on the drilling environment, depth of the bore and the properties of the soil. They are;
Auger drilling
Air core drilling
Cable tool drilling
Diamond core drilling
Hydraulic-rotary drilling
Sonic (vibratory) drilling / Resonance enhanced drilling
Reference [2][3]
2.2 Resonance enhanced drilling (RED)
Above first 5 methods are conventional drilling techniques, they require high power and high torque for drilling therefore they can be both expensive and time consuming. But sonic drilling (resonance enhanced drilling) provides a controllable dynamic loading. This variable load can be useful in different drilling conditions and it makes it possible to drill through in different formations with the same drill tool. The figure 3 shows the simplified idea of resonance enhanced drilling.
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Figure 3: With and without Resonance drilling as shown at [6]
In this project we focus on resonance enhanced drilling and manufactured experiment rig can be assumed as an example of a resonance enhanced drilling system.
Reference [5][6]
2.3 Hydro dynamic forces on the drilling tool
Fluid hydrodynamic forces due to the presentence of water in the borehole affects the dynamic motion of the drilling machine, especially the drilling tool and hence it's necessary to quantify mechanical properties such as density of the fluid, viscosity and the frequency of the oscillation of water and it's damping effect.
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Figure 4: Borehole and the drilling tool diagram as shown at [8]
Boreholes are deeply penetrated holes in to the earth to deliver oil from a reservoir as shown in the figure 4. During borehole drilling, under the action of high pressure the drilling mud occupies oil concentrated formation. The drilling tool displaces the fluid mixture which is called pore fluid. The filtering process and the drilling process is rapidly slow down by the growth of mud on the borehole wall surfaces. This is due to increasing in viscosity of the fluid and its high damping effect.
In this project, an experimental rig has been manufactured to determine the fluid hydrodynamic forces by using cylinder floating in a fluid as shown in Figure 1. The larger cylinder (bore cylinder) refers to the borehole, which goes in to the earth. The small cylinder refers to a part of the drilling tool.
Reference [2][3][6][7]
2.4 Equipments arrangement
Sinusoidal forces are generated on top of the system and delivered to the smaller cylinder. A function generator, which is connected to the vibro-impactor, varies the frequency of the oscillation. The amplitude of the oscillation is varied by a transformer by varying the voltage. Voltmeter shows the variable voltage. Simple block diagram below shows how the signal input flows to the system.
Figure 5: Block diagram - Signal input
All the transducers are attached to the experimental rig in appropriate positions one at a time or two together. The oscilloscope and the computer are connected to the transducers in order to record the experimental results.
Figure 6: Block diagram - Signal output
Arrangement of the entire experiment rig can be simplified as shown in figure 7
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Figure 7: Arrangement of the equipments
3.0 Equipment
3.1 Motor selection
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Figure 8: Eccentric mass assembly of the motor as shown at [9]
A motor can be used to generate sinusoidal forces in the experiment rig by connecting an eccentric mass as shown in the figure 8. Main aspects of the motor are listed below.
Weight of the motor has to be low.
It's easier to supply power for a DC motor.
Not necessary to have a large stall torque.
But motor with an eccentric mass can create vibrations not only vertically but also horizontally. This can create unnecessary friction in the vibrating system.
3.2 VIbro - Impactor
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Figure 9: Vibro Impact mechanism as shown at [11]
This device is the substitution for the electric motor and it creates vertical sinusoidal forces to the vibrating system. The solenoid creates a magnetic flux according to the current supplied. This large electromagnetic force pulls the metal bar loaded with masses towards the centre of the solenoid with the help of the gravity. Due to the tensions of the top springs and the inertia of the metal bar and the attached masses move past centre of the solenoid and reaches the maximum height. When the metal bar is at the maximum point, direction of the supplied current is changed. Therefore the direction of the induced magnetic flux is opposite. Then magnetic force will pull the metal bar attached with masses as before. Force acting on the metal bar can be calculated by using the equations below.
(1)
Whew F = force, v = instantaneous velocity of the metal bar, q = electric charge, B = magnetic field given by;
(2)
Where µ = permeability of the metal core, N = number of loops, L = length of the wire, I = current
This is a very simple design and this is the most appropriate for this experiment as it does not vibrate horizontally and it's easier to supply a variable signal than an electric motor. A system of vibro-impact can have various engineering application. Example ground moling and pile driving.
Reference [10]
Pressure transducer selection
Figure 10: Pressure transducer as shown at [13]
Pressure transducer / sensor can be placed at the bottom of the cylinder to measure the variable pressure of the fluid due to the floating mass. This equipment generates an electric signal as a function of the pressure imposed. By connecting the pressure transducer to an oscilloscope or to a computer software the nature of the waves can be examined. It also can measure the water level in the container, pressure realising speed and the altitude by providing the necessary data to it.
Pressure sensing and fluid level sensing are the necessary objectives of the pressure transducer in this experiment. Fluid level can be calculated by taking the proportionality between fluid level and the pressure at the bottom. A basic equation for such a measurement is,
P = hï²g (3)
Where P is pressure at the bottom of the container, Ï is the fluid density, g acceleration of free fall and finally h is the height of fluid column above the pressure senor. Pressure transducer still needs to be purchased. All suitable pressure transducers are listed in the appendix, which are found in RS Components Sdn Bhd.
Pressure transducer 1 in appendix is the most suitable for this experiment.
Reference [12]
Displacement Transducer selection
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Figure 11: Linear variable displacement transducer (LVDT) as shown at [15]
This device is also called as the Linear Variable Displacement Transducer (LVDT). Oscillating amplitude of the inner cylinder is measured by using a displacement transducer.
This device uses the principle of Eddy-current measurement by moving an aluminium rod along the steel cylindrical housing. Coils will detect the changing the induced Eddy-current, thus changing the coil impedance. An electronic circuit in the transducer head transforms the information of the measuring rod position into a linear signal. The sensor is actively compensated for temperature changes.
This can be connected to either computer or to oscilloscope to record data. Computer software is always preferable since it can filter better than the oscilloscope and it can recode data easily. However oscilloscope can show the direction and generalised motion of the system.
The available LVDT in the lab is Solartron DFG 2.5 (Part number - M922938A443-03) from RS.
Reference [14]
Accelerometer selection
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Figure 12: Section view of an accelerometer as shown at [17]
The acceleration of the inner cylinder can be measured by using an accelerometer. The accuracy of this equipment is sufficient enough and it can provide data to an oscilloscope or to a computer software.
Single- and multi-axis models are the main two types available in the market. This unit is always provided with the driver and the software so that it can directly connect to a computer and analyse the behaviour of the vibration, mainly with graphs. Another advantage of this is equipment is, its easer to fix in the experimental rig than the LVDT.
In this experiment we use an accelerometer to measure the acceleration of the vibrating system as a function of time. KISTLER Ceramic Shear Accelerometer (Part number - 8877650M6) is the one used in this experiment.
Reference [16]
4.0 Manufacturing of the components
The entire experiment rig is manufactured in the university laboratory and it's made in a way that we can adjust all the parameters at any time we needed. This is done due to the rapid chain in structure of the rig. Even the manufacturing materials are; easily found in the workshop and they can be easily manufactured.
4.1 Bore cylinder
The smaller cylinder needs to be visible while it's floating on water inside the bore cylinder, therefore this unit is manufactured with Perspex due to its transparency, inexpensiveness and mainly manufacturing methods are much easier when compared to glass. This hollow cylinder and the bottom plate ate joined together by using water-proof glue.
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Figure 13: Bore Cylinder from the experiment rig
4.2 Two plates on top of the Bore cylinder
These 2 plates are cut with Perspex and resized in to proper dimensions by using the milling machine. The centre of the plate is marked with the edge finder tool and drilled through with a 26mm drilling tool. Both Perspex plates are attached together with 2 small Teflon pieces in between them as shown in the figure 14. Teflon pieces are resized in to proper dimensions and all the parts are joined together with 4 x M3 bolts and nuts.
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Figure 14: 2 x Perspex plates
Inner cylinder and the connecting rod
Initially we were supposed to manufacture the inner cylinder with an Aluminium hollow cylinder. But the current arrangement of the rig is with a tin can as shown in the Figure 15. Weight of the Aluminium cylinder is the main reason to switch to the tin can. The tin can doesn't need any additional manufacturing processes other than just drilling 2 holes across it to hang it to the PVC tube as shown in figure 15.
Even a PVC tube is used due to the light weight and less manufacturing process.
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Figure 15: Inner tin can and the PVC tube
4.4 Aluminium casing for the Vibro-Impactor
This is made out of 4 Aluminium sheets cut in to proper dimensions joined together with 4 x M4 bolts as shown in the Figure 16. There is threaded hole in the middle of the top plate to place the accelerometer. Mainly milling machine and folding machine are used to manufacture these aluminium parts.
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Figure 16: Aluminium casing and the Vibro-Impacter
5.0 Mathematical model for the experimental results
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Figure 17: Theoretical analysis of the system
The experimental rig is compared with the simple spring mass damper system as shown in the right side of the Figure 17.
According to Newton's first law, if a system is vibrating with its natural frequency and if there is no external force applied on the system, then the system will continue to vibrate in the same frequency forever. But in reality, in this experiment the fluid will act as a damper for the oscillating system as shown in the diagram. So the sinusoidal force needs to be applied to the top of the system continuously to determine necessary data. This data is recorded by using an accelerometer and a linear displacement transducer which are attached to the system together or separately.
While the system is oscillating accelerometer will measure the acceleration of it as function of time and the LVDT will measure the displacement (amplitude) as a function of time. Pressure transducer could be used to measure the variation of pressure at the bottom of the cylinder and change in fluid level due to the oscillation.
Reference [1][10][18][19][20][21][22]
5.1 Data Acquisition
The experiment is carried out in various types of cycles and imported data to the computer. They are, fixing all the available transducers, by fixing either accelerometer or displacement transducer, collecting data of a continuously forced oscillating system, collecting data of a transient system by supplying an impulse to it and all these data are available in 2 sets and they are taken with water in the bore cylinder and without water in it.
The transient motion data within a chosen region of the linear variable displacement transducer (LVDT) with water is graphically represented below.
Water damped system with the LVDT: Displacement vs. Time
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Figure 18: Displacement vs. time (Data form LVDT)
The transient motion data within a chosen range of the accelerometer with water is graphically represented below. From experimental data obtained from Mr. Khoo Wen Jia, the following graphs are plotted.
Water damped system: Acceleration Vs. Time
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Figure 19: Acceleration vs. Time with water (Data from accelerometer)
For small amplitude and small frequency oscillations, the vibration of the system can be assumed as linear. The sinusoidal curvature becomes non linear at higher amplitudes (Higher voltages more than 12v). This is due to higher friction and the in alignment of the system.
Since this is a sinusoidal transient curve within the given region and the motion of the total vibrating system with water is assumed as simple harmonic. Therefore the function of the above acceleration vs. time graph (Figure 19) can be assumed as,
(4)
Where a, b, c are numerical constants and t is time.
Velocity of the oscillating system obtained by integrating the equation 4,
(5)
The velocity curvature of the above equation (equation 5) is shown below.
Water damped system: Velocity vs. Time
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Figure 20: Velocity vs. Time with water (Data from equation 5)
Displacement (amplitude) of the oscillating system is obtained by integrating the equation 5,
(6)
Displacement curvature of the above equation (equation 6) is shown below.
Water damped system: Displacement vs. Time
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Figure 21: Displacement vs. Time with water (Data from equation 6)
Theoretical modal for the experiment
Theoretical model could be used to compare and match with the experimental data. This can be useful to evaluate the numerical constants in the equation 4, 5, 6. Especially the damping coefficient "b" could be evaluated in the above equations by this method. Method of multiple scale analysis is used to create a theoretical model.
6.1 Multiple scale analysis
Multiple scale analysis is a global perturbation method and used to construct uniformly valid approximations to the solutions of perturbation problems. This method is valid for both for small as well as large values of the independent variables. This is done in 2 different ways, its introducing a fast scale or a slow scale to the problem.
Multiple scale methods and standard equations
Many variable version (The derivative expansion procedure).
The two variable expansion procedure.
Generalized method - Non linear scale.
The Duffing equation.
The Van Der Pol oscillator
(7)
(8)
The general form of the equation was developed over several years. Main investigators are Kuzmak (1959), Cochan (1962), Mahony (1962) and Nayfeh (1965). In orbital mechanics, Nayfeh used the generalized version of multiple scale analysis to investigate the earth moon spaceship problems. It's a general study of taking off a satellite from a circular orbit and an elliptical orbit with a small thrust. Kevorkian and Brofman studied the motion of a satellite under a minor thrust or drag. These investigators studied further about behaviour of satellites and obtained nonlinear dynamic equations.
Multiple scale method is further discovered in water waves, gravity waves in water of variable depth, shallow water waves in shear flows, free surface oscillation in a water tank etc.
To compare the experimental results with a theoretical method, It's much appropriate to use Many variable version. The Duffing equation is used to obtain the results since it gives much better results with water vibrating systems and its very appropriate for low amplitude oscillations.
Reference [23]
The Duffing Equation is modified as it matches the experiment nature. Which is a transient oscillation and it needs to contain a damping term of .
Duffing equation with the damping term can be written as below;
(9)
Assume that;
(10)
Standard substitutions for multiple scale analysis are shown in the equation 11 and equation 12
(11)
And;
(12)
Substituting equation (11) and (12) in equation (9) and equating coefficients of each power of ϵ to zero, we have; (Refer appendix for complete simplification)
(13)
(14)
(15)
Solution for equation (13) is;
(16)
By substituting the equation (16) in to (14), then equation (14) will become [23];
(17)
Positive power of the exponential will create an oscillation with increasing amplitude as shown in the diagram below.
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Figure 22: Graph of a positive coefficient of an exponent
This sort of motion is practically impossible for a damping system. Therefore it can be equal to zero.
(18)
Therefore equation 17 can be written as follows;
(19)
Then u1, , will be;
(20)
(21)
(22)
Equation (20) and (22) can be substituted to equation (19_;
(23)
(24)
(25)
According to the [23] the expression for u1 will be;
(26)
Similarly from [23] u2 will be;
(27)
According to the book expression for u can be written as below;
(28)
Where
(29)
The exact solution for the Duffing equation with the damping coefficient is not available. There for the above equation (28) cannot be compared with the experiment results.
The results could be plotted on the same set of axis as acceleration vs. time, velocity vs. time and finally displacement vs. time as shown an example below.
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Figure 23: Example of compression of the theatrical and experimental values
Reference [23][24][25][26]
7.0 Conclusion
The experimental readings show a linear motion of the vibrating system within certain range of frequency and voltage. The smoothness of the graph will not be the same when the system is vibrating at high frequency and with high amplitudes. This is mainly because friction of the rubbing parts due to nonalignment of the system, the noise in the surrounding and hanging wires of the system as shown in Figure 24.
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Figure 24: Current top arrangement of the experiment rig
The entire structure of the experiment rig is changed several times. Therefore manufacturing of components are not highly accurate. Some parts of the experiment rig are attached with rubber bands, these needs to be modified and get rid rubber bands because rubber bands effect the vibration of the system.
The entire experiment rig needs to be designed on a cad software accurately. The new design should contain a stool to hold the bore cylinder. This will allow to have space for the pressure transducer which needs to be placed at the bottom of the cylinder as shown in the figure 25. Remanufacturing the new design will reduce the problems that I mentioned above.
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Figure 25: Stool for the bore cylinder to place the pressure transducer
More experimental data could have been taken if a pressure transducer is available. The results from the pressure transducer could directly added to the equation (3) and evaluate the variation of the water height. This could be compared with the LVDT values of each experiment.
The theoretical model could be used for comparison with all the experimental results. The damping ratio "b" of the fluid inside the bore cylinder can be determined by using equation 4, 5, 6. But the damping coefficient of a fluid is not constant every time. Velocity of the fluid isn't the variable for a damping coefficient. It's possible to reduce the other factors to a constant and determine the Reynolds number which is a function of viscosity and the geometry of the inner cylinder as shown in the equation 30.
(30)
Where, Ï is the fluid density. V =mean fluid velocity, L = characteristic linear dimension, µ = is dynamic viscosity.
Damping effect will be directly proportional to the viscosity of the fluid as shown below.
(31)
Where C is the damping coefficient and µ = is dynamic viscosity.
8.0 Reference
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