You are an expert at summarizing long articles. Proceed to summarize the following text:

You are an expert at summarizing long articles. Proceed to summarize the following text: 
BACKGROUND OF THE INVENTION 
       [0001]    The present invention relates to a method for fracturing the earth from a wellbore by over pressuring a fluid(s) and/or gases inside a wellbore under conditions of resonance. 
       BRIEF DESCRIPTION OF PRIOR ART 
       [0002]    Fracturing the earth from a wellbore is a known technique for enhancing oil production and recovery from an oil bearing bed. A variety of methods and apparatuses have been proposed to create both short and long fractures near a wellbore. One of these is described and claimed in U.S. Pat. No. 5,617,921 by Schmidt et al., herein incorporated by reference, wherein a method for initiating and/or extending a fracture in an earth&#39;s formation from a well penetrating the formation utilizing a source of high pressure fracturing fluid disposed on the earth&#39;s surface which is released to flow into and through the well at a predetermined time to initiate and/or extend the fracture. However, this method requires a significant amount of energy and to have a relatively large diameter tubing string in which to hold a sufficient charge of pressured gas to provide an adequate fracture fluid pressure and flow characteristics. 
         [0003]    The use of high pressured gas or other pressured fluid(s) in a wellbore to clean perforations and/or create fractures has been described in U.S. Pat. Nos. 5,669,448 and 5,131,472, herein incorporated by reference. These references disclose a system of stimulating a well by suddenly applying pressure to the formation in excess of the fracture gradient pressure and thereafter pumping fluid into the well before the pressure declines below the fracture gradient pressure. In addition, there are other more expensive means of injecting treatment liquids that have been proposed. One such type of approach is to place the treatment liquid in the well and ignite a gas generating propellant in the production string, as shown in U.S. Pat. Nos. 6,138,753; 5,443,123; 5,145,013; 5,101,900; 4,936,385 and 2,740,478, herein incorporated by reference. Of more general interest is the disclosure in U.S. Pat. No. 3,029,732, herein incorporated by reference. 
         [0004]    While there have been a variety of methods proposed for creating hydraulic fractures around the wellbore, there remains a need for an effective, high-pressure method which creates a pattern of fractures extending from all perforations into the formation in particular with the required parameters of fractures. 
       SUMMARY OF THE INVENTION 
       [0005]    Accordingly, a primary object of the present invention is to provide an apparatus for enhancing of forming at least one fracture having a required width r and length L in the formation surrounding a wellbore in the regime of resonance which includes a flow line at the surface supplying a fracturing fluid from a tank via a pump into the wellbore and a flow line having a check valve preventing flow of fracturing fluid from the wellbore back into the flow line, a tubing string connected to the flow line and extending downwardly into the wellbore, an elongated cylinder connected to the bottom of tubing string at the upper end and having an opening to wellbore, a plunger movably arranged within an elongated cylinder to move within the elongated cylinder and having a taper on a lower end of said plunger, the pumping means, accommodated inside a lubricator to prevent the leakage of the fracturing fluid from the tubing and the flow line at the surface, and connected with plunger for moving of said plunger within the elongated cylinder and compressing the fracturing fluid contained between check valve inside the flow line and plunger inside the elongated cylinder and discharging said fracturing fluid into the wellbore via an opening when said plunger exits out of the elongated cylinder on every upstroke of pumping means to generate a vibrations on resonant frequency, and an angle θ of taper on lower end of said plunger is determined by the following expression: 
         [0000]    
       
         
           
             
               θ 
               = 
               
                 
                   1 
                   3 
                 
                  
                 arc 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 
                   ine 
                    
                   
                     [ 
                     
                       - 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               ϕ 
                             
                             ) 
                           
                            
                           
                             
                               ShL 
                               s 
                             
                              
                             
                               ( 
                               
                                 
                                   D 
                                   1 
                                   2 
                                 
                                 - 
                                 
                                   d 
                                   r 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             f 
                             r 
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           
                             tD 
                             1 
                             3 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where θ is the angle of taper on lower end of the plunger, φ is a slippage of fracturing fluid between the elongated cylinder and plunger, Sh is Strouhal number, L s  is the length of stroke of the pumping means, D 1  is the diameter of plunger, d r  is the diameter of pumping means, f r  is the resonant frequency, Δt is the discharging time of compressed liquid. 
         [0006]    It is another object of the present invention to provide an apparatus for enhancing of the formation fracturing around a wellbore in the regime of resonance in which the length l of said taper on the lower end of plunger is determined by the following expression: 
         [0000]    
       
         
           
             
               
                 0.1 
                  
                 
                     
                 
                  
                 
                   D 
                   1 
                 
               
               ≤ 
               l 
               ≤ 
               
                 
                   D 
                   1 
                 
                 
                   2 
                    
                   
                       
                   
                    
                   tg 
                    
                   
                       
                   
                    
                   θ 
                 
               
             
             , 
           
         
       
     
         [0000]    where l is length of taper on the lower end of plunger, D t  is the diameter of plunger, θ is the angle of taper on lower end of the plunger. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]      FIG. 1  shows a schematic illustration of the wellbore in which the method and the apparatus of the present invention is employed. 
           [0008]      FIG. 2  is a cross-sectional side view of the apparatus for practicing the present invention. 
           [0009]      FIG. 3  is a cross-sectional top view of the wellbore and the formation with the fractures. 
           [0010]      FIG. 4  is a waveform and spectrum of a single burst/pulse provided by device for generating vibrations. 
       
    
    
     DETAILED DESCRIPTION 
       [0011]    Referring to  FIG. 1  and  FIG. 2 , there is shown the wellbore  1  having perforations  5  and fractures  6 .  FIG. 1  shows a general arrangement of a device for generating vibrations and procedure using the vibrations, the flow line  11  at the surface supplying the pressurized fracturing fluid from tank  13  via pump  12  into wellbore  1 , the check valve  10  which is closed when the pressure of fracturing fluid inside the tubing  2  is greater than the one in flow line  11  thereby preventing flow of fracturing fluid from the tubing  2  back into the flow line  11 , the tubing string  2  connected to flow line  11  and extending downwardly into the wellbore  1 , a device for generating vibrations consisting of the elongated cylinder  3  connected with the bottom of tubing string  2  at the upper end and having the opening  8  to wellbore  1 , the plunger  4  having the taper  16  movably arranged within the elongated cylinder  3  to move within said elongated cylinder  3 , the pumping means  7  (for instance, sucker rods which, in turn, are connected to pumping unit) connected with the plunger  4  for moving the plunger  4  within the elongated cylinder  3  and compressing the fracturing fluid contained between the check valve  10  inside the flow line  11  and plunger  4  inside the elongated cylinder  3  and discharging the compressed fracturing fluid into the wellbore  1  via the opening  8  when plunger  4  exits out of the elongated cylinder  3  on every upstroke of the pumping means  7  to generate the vibrations due to the creating a periodic vortices  14  ( FIG. 2 ) in accordance with well known phenomenon of an auto-oscillations discovered by V. Strouhal in 19 th  century. A lubricator  9  accommodates a pumping means  7  to prevent the leakage of the compressed fracturing fluid from the tubing string  2  and flow line  11  at the surface. More details about phenomenon of auto-oscillations could be found for example in the articles: Sobey, Ian J. (1982). “Oscillatory flows at intermediate Strouhal number in asymmetry channels”. Journal of Fluid Mechanics, N. 125: 359-373, herein incorporated by reference, and Sakamoto, H.; Haniu, H. (1990). “A study on vortex shedding from spheres in uniform flow”. Journal of Fluids Engineering, N 112 (December 1992): 386-392, herein incorporated by reference. 
         [0012]    The generation of vibrations is repetitive and occurs in the form of bursts or so called hydro-impact waves/pulses at the moment when plunger  4  exits out of the elongated cylinder  3  due to the upward motion of the pumping means  7 . The pumping means  7  provide a reciprocating upward and downward movements of the plunger  4  inside the elongated cylinder  3 . The number or rate of reciprocating movements may vary from a few per hour to a dozens per minute depending on the particular fracturing operation. 
         [0013]    The typical measured waveform of a single burst and corresponding spectrum are shown on  FIG. 4 . The diagram/waveform on the left part of  FIG. 4  shows the signal from device for generating vibrations recorded by a geophone and two geophones, namely, horizontal and vertical ones in the offset well located on the distance of 1385 feet from the well wherein the device for generating vibrations was installed. The duration of the bursts accounts for 40-100 milliseconds depending on pressure of the compressed fracturing fluid between the plunger  4  and the check valve  10 . A main or resonance frequency of generated vibrations can be “moved” along the frequency axis to the left or right on the spectrum diagram by providing the device for generating vibrations having an ability to create vibrations on a particular resonant frequency, i.e. the frequency matching so called eigen frequency of fractures with predetermined or required width r and length L ( FIG. 3 ). As it is seen from  FIG. 4  the amplitude of the resonant frequency is by about 30-50 times higher compared with the rest of frequencies in spectrum (the units on vertical axis are in decibels). The check valve  10  installed on a flow line  11  could have a simple design having a seat with round hole in the center of said seat and a ball having bigger diameter and matching said hole in such manner that when the pressure of fracturing fluid in front of ball is greater than behind the one the ball closes said round hole of said seat thereby preventing any backward flow of fracturing fluid from flow line  11  into the tank  13 . It should be noted that valve  18  during fracturing has to be either closed or at least one standard packer (not shown) is installed between tubing  2  and casing  15  above the perforations  5 . 
         [0014]      FIG. 3  shows the cross-sectional top view of the wellbore  1 , a casing  15 , cement bond  17 , and the formation with the perforations  5  and the fractures  6 . The eigen, natural or resonant frequency of such fractures (or slots in acoustics) is determined by the following formulae: 
         [0000]    
       
         
           
             
               
                 f 
                 r 
               
               = 
               
                 
                   c 
                   
                     2 
                      
                     
                         
                     
                      
                     π 
                   
                 
                  
                 
                   
                     r 
                     
                       1.2 
                        
                       
                           
                       
                        
                       
                         HL 
                          
                         
                           ( 
                           
                             r 
                             + 
                             W 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where f r  is the frequency of resonance, c is a speed of sound in the fracturing fluid, π equals 3.1415, r is the required width of fracture  6 , H is the combined thickness of the casing  15  and the cement bond  17  surrounding the casing, L is the required length of fracture  6 , W is a length of the a casing arch between two neighboring perforations  5 . In particularly, for fracturing event shown on  FIG. 3  (four fractures  6 ) W=πD/4, where D is the diameter of the casing  15 . Thus, in order to get the fracture(s)  6  with particular parameters, i.e. the required width r and length L, the affecting vibrations have to be supplied on corresponding resonant frequency. For instance for the following parameters: r=0.02 m, H=0.05 m, L=100 m, W=0.13 m (corresponds to 7.0 inch casing and four fractures), c=1600 msec (corresponds to 70 MPa hydrostatic pressure under 20° C. temperature in wellbore) the resonant frequency equals 38 Hz. It should be noted that under conditions of resonance the fractures  6  will have predetermined, required width r and length L. It&#39;s important from the point of view of fracturing process when in case of too wide width r the excessive usage of proppant could be prohibitively expensive, and the fracture having too long length L can reach the highly water saturated sublayer of formation leading to the production of excessive portion of water instead of oil or gas from fractured well. 
         [0015]    To generate the vibrations on resonant frequency the device for generation of vibrations in accordance with present invention has to have the corresponding angle θ of taper  16  ( FIG. 2 ) on the lower end of said device determined in accordance with the following formulae: 
         [0000]    
       
         
           
             
               θ 
               = 
               
                 
                   1 
                   3 
                 
                  
                 arc 
                  
                 
                     
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 
                   ine 
                    
                   
                     [ 
                     
                       - 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               ϕ 
                             
                             ) 
                           
                            
                           
                             
                               ShL 
                               s 
                             
                              
                             
                               ( 
                               
                                 
                                   D 
                                   1 
                                   2 
                                 
                                 - 
                                 
                                   d 
                                   r 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         
                           2 
                            
                           
                               
                           
                            
                           
                             f 
                             r 
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           
                             tD 
                             1 
                             3 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where θ is the angle of taper on lower end of the plunger, φ is a slippage of the fracturing fluid between the elongated cylinder and plunger (no dimensional), Sh is Strouhal number, L s  is the length of stroke of the pumping means, D t  is the diameter of plunger, d r  is the diameter of pumping means, f r  is the resonant frequency, Δt is the discharging time of compressed liquid. 
         [0016]    For instance for the following parameters: D 1 =0.08255 m, φ=0.1, Sh=0.21, d r =0.0254 m, L s =2.8 m, Δt=0.25 sec and resonant frequency f r =38 Hz the corresponding angle θ of taper  16  equals 36°, meanwhile the length l of taper  16  on the lower end of plunger  4  is determined by the following expression: 
         [0000]    
       
         
           
             
               
                 0.1 
                  
                 
                     
                 
                  
                 
                   D 
                   1 
                 
               
               ≤ 
               l 
               ≤ 
               
                 
                   D 
                   1 
                 
                 
                   2 
                    
                   
                       
                   
                    
                   tg 
                    
                   
                       
                   
                    
                   θ 
                 
               
             
             , 
           
         
       
     
         [0000]    where l is length of taper on the lower end of plunger, D t  is the diameter of plunger, θ is the angle of taper on lower end of the plunger. 
         [0017]    Thus for above noted parameters (D 1 =0.08255 m and θ=36°) the length l of taper  16  can vary in the range of 0.008255 m to 0.05681 m depending on particular design of elongated cylinder  3 . 
         [0018]    As an alternative source of the said vortices generating vibrations on resonant frequency, for instance, can be used the devices described in U.S. Pat. No. 8,459,351 U.S. Pat. No. 6,015,010 and U.S. Pat. No. 6,899,175, herein incorporated by reference. 
         [0019]    While in accordance with the provisions of the Patent Statutes the preferred forms and the embodiments of the invention have been illustrated and described, it will be apparent to those of ordinary skill in the art various changes and modifications may be made without deviating from the inventive concepts set forth above.

Summary:
An apparatus for generating of vibrations for enhancing of forming at least one fracture having a required width r and length L in the formation surrounding a wellbore in the regime of resonance by means of applying vibrations to the formation which is undergoing pressurizing exceeding the fracture gradient pressure of the formation.