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 
     1. Field of the Invention 
     The present invention relates generally to sucker rod strings, and more particularly to a composite sucker rod string that utilizes both fiberglass reinforced, resin bonded sucker rods and steel sucker rods for connecting a subsurface well pump to a surface mounted pumping unit. 
     2. Description of the Prior Art 
     It is well known to utilize a plurality of sucker rods to form a sucker rod string for connecting a subsurface well pump to a surface pumping unit in order to impart a reciprocating pumping motion to the subsurface well pump. The sucker rods forming the sucker rod string have generally been fabricated from steel. Such steel rods have produced adequate pumping action; however, problems have been encountered with steel sucker rods in wells having heavy pumping loads, for example, as encountered in wells having low fluid levels and in deep wells. In such a well, the weight of an all-steel sucker rod string combined with the weight of the fluid load imposes an undue load on the surface pumping unit. Moreover, the life of an all-steel sucker rod string is limited when such a string is used in a corrosive well, since the corrosive action of the well tends to corrode and ultimately weaken the sucker rod string to the breaking point. 
     Recently, polyfilament reinforced, resin bonded sucker rods utilizing, for example, fiberglass reinforcing fibers have been introduced. Such polyfilament fiberglass sucker rods are described in U.S. patent application Ser. No. 576,731, filed May 12, 1975, and in U.S. patent application Ser. No. 956,740, entitled &#34;IMPROVED SUCKER ROD AND INTERCONNECTIONS THEREFOR&#34;, filed concurrently by the present inventor, and assigned to the same assignee. Both applications are incorporated herein by reference. Such fiberglass sucker rods have the advantage that they are substantially lighter than steel, and that they are noncorrosive. Thus, such fiberglass rods reduce the load on the surface pumping unit, and they do not corrode in corrosive wells. However, fiberglass sucker rods have two properties that were heretofore considered to be disadvantages. They cannot support great compressive loads, and their modulus of elasticity is less than that of steel rods. Thus, it has been customary practice to utilize heavy steel bars known as sinker bars at the bottom end of the string in order to preload the string and maintain the fiberglass sucker rods in tension during the entire pumping stroke. However, in such a system, the reduced modulus of elasticity of the fiberglass rods causes fiberglass rods to stretch more than steel rods during the pumping cycle, and such stretch has caused reduced pumping efficiency by reducing the pump stroke imparted to the subsurface well pump. The present invention makes use of the increased stretch of the fiberglass rods, previously considered to be a disadvantage, to actually increase the pump stroke over that which would be obtained with relatively inelastic steel rods. 
     SUMMARY OF THE INVENTION 
     Accordingly, it is an object of the present invention to provide an improved sucker rod string utilizing relatively elastic sucker rods that overcomes many of the disadvantages of the prior art sucker rod strings. 
     It is another object of the present invention to provide an improved sucker rod string that combines the advantages of relatively elastic and relatively inelastic sucker rods. 
     It is another object of the present invention to provide an improved sucker rod string that combines the advantages of steel and polyfilament sucker rods. 
     It is yet another object of the invention to provide an improved sucker rod string that utilizes the inherent elasticity of polyfilament sucker rods, such as, fiberglass reinforced, resin bonded sucker rods to increase pumping efficiency. 
     It is still another object of the present invention to provide an improved sucker rod string that provides greater pumping efficiency than the prior art sucker rod strings. 
     It is still another object of the present invention to provide an improved sucker rod string that provides greater pumping capacity while simultaneously reducing the load applied to the surface pumping unit. 
     In accordance with a preferred embodiment of the invention, a sucker rod string is fabricated from a plurality of polyfilament reinforced sucker rods and a plurality of steel rods. In the present embodiment, continuous filaments of fiberglass which make up on the order of 79-80% of the rod by weight are used in the polyfilament rods, but other materials, such as graphite may be used in the fabrication of the polyfilament rods. The relative lengths of the polyfilament and steel portions of the sucker rod string are determined by the various well parameters, such as pump diameter, pump stroke, surface unit stroke, number of strokes per minute in order to optimize pumping efficiency. Such optimization often results in a pump stroke at the subsurface well pump that is greater than the stroke of the surface unit, thus greatly increasing production. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     These and other objects and advantages of the present invention will be readily apparent upon consideration of the following detailed description and attached drawing, wherein: 
     FIG. 1 is a plan view of a sucker rod string according to the invention connecting a pumping unit and a subsurface well pump; and 
     FIG. 2 is a graphical representation showing the effects of stretch in the sucker rod string during various portions of the pumping cycle. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring now to the drawing, with particular attention to FIG. 1, there is shown a surface pumping unit 10 that actuates a subsurface pump 12 by means of a sucker rod string 14 which comprises several relatively elastic polyfilament fiberglass reinforced, resin bonded sucker rods 16 connected end-to-end and a plurality of similarly connected relatively inelastic steel sucker rods 18. In a typical embodiment, the pump 12 may be several thousand feet below the surface and the sucker rod string may include several hundred rods 16 and 18. In a preferred embodiment, each of the polyfilament rods 16 is approximately 37.5 feet in length and has a diameter of approximately 7/8 inch or, more precisely, 0.845 inch. The percentage of fiberglass content in such rods is on the order of 79-80% by weight, and the resin is polyester. The steel sucker rods 18 are typically 25 feet in length and have a similar diameter. However, the relatively elastic and relatively inelastic rods may be fabricated in various lengths and diameters, and fabricated from various materials. Also, more than two different types of rods may be used, particularly if a more complex weight and elasticity distribution is required. 
     When a sucker rod string of the type illustrated in FIG. 1 is utilized, the dynamics of the pumping action results in a periodic stretching of the string 14, particularly of the fiberglass portion. Such stretching was previously considered to be disadvantageous; however, it has been found that by properly adjusting the elasticity of the sucker rod string by adjusting the length of the relatively elastic fiberglass portion of the sucker rod string relative to the length of the steel portion, as well as the pumping speed, the length of the stroke at the subsurface pump 12 can be made to exceed the length of the stroke produced by the surface pumping unit 10. Such an increase in stroke provides a substantial increase in the amount of fluid that can be pumped in a given time interval compared to that which can be pumped by relatively inextensible steel sucker rod strings within the same time period. The dynamics are illustrated in Diagrams I-IX of FIG. 2. 
     Basically, the increased pump stroke S p  occurs as a result of variations in the stretch of the sucker rod string during the pumping cycle, particularly of the fiberglass portion which is more elastic than the steel portion. Such variations in the stretch of the rod string occur as a result of changes in the pump load as a result of the weight of the fluid pumped, and as a result of the acceleration and deceleration of the sucker rod string, particularly of the steel portion, when its direction of travel is changed. For example, when the sucker rod string is stationary or travelling in a downward direction at a uniform rate, as illustrated in Diagram I (FIG. 2), and the string is not being subjected to any fluid load forces or accelerations other than gravity, the only force stretching the fiberglass string is its own weight and the weight of the steel string. Similarly, the only force stretching the steel string is its own weight. However, when the arm of the pumping unit 10 reaches the bottom of its pumping stroke and instantaneously stops prior to beginning its upward stroke, as illustrated in Diagram II (FIG. 2), the bottom of the rod string does not stop instantaneously, but rather, continues to move downward as a result of inertia (i.e., the property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force), particularly the inertia of the steel rods 18. The continued movement causes the fiberglass portion of the string to stretch by an amount OT, which results in overtravel at the bottom of the rod string. As the surface pumping unit 12 begins its upward stroke (Diagram III, FIG. 2), the fiberglass portion of the string is further stretched as the fiberglass string accelerates both the steel string and the fluid load of the pump in an upward direction. The rod string must also accelerate the fluid in the pump in an upward direction, and the contribution to the overall stretch produced by the fluid load is represented by the term RS (Diagram IV). Thus, the upper end of the string rises faster than the lower end. The steel portion of the sucker rod string is also stretched in a similar manner; however, since the major portion of the stretch occurs in the fiberglass portion of the string, a discussion of the effects of the steel portion will be deferred until a subsequent portion of the specification for reasons of clarity. 
     In a similar fashion, as the pumping unit approaches the top of its stroke, the inertia in the string, as well as the energy stored in the string in the form of rod stretch, cause the lower end of the string to continue to move in an upward direction, thereby reducing the amount of stretch in the fiberglass portion of the rod (Diagrams IV and V, FIG. 2), until the amount of stretch is reduced to its static value or less (Diagram VI) whereupon the cycle is repeated (Diagrams VII-IX). 
     In order to obtain an increase (rather than a decrease) in the pump stroke at the bottom of the well, the rod string must be tailored to operate within the constraints of a particular well, and its parameters must be adjusted to conform to various well parameters, such as depth, fluid load, and extraneous factors such as friction. In particular, the weight of the fluid lifted during each pump stroke, as well as the weight of the rod string and the elasticity of the rod string, must be considered. This is because, under dynamic conditions, the weight of the rod string periodically stretches the rod string and thus stores potential energy in the rod string during a portion of each pumping cycle (Diagram III). The potential energy thus stored can then be retrieved during another portion of the pumping cycle to aid in the lifting of the fluid (Diagrams IV and V). If the force resulting from the stored potential energy when the rod subsequently contracts is greater than the force required to lift the fluid load during the up stroke of the pump, a net increase in pump stroke results. Conversely, if insufficient potential energy is stored as a result of too heavy a fluid load, or too elastic or too light a rod string, the net pump stroke is actually decreased. 
     The pump stroke S p  is related to the surface stroke, rod weight and fluid load according to the following relation: 
     
         S.sub.p =S+OT-RS                                           (1) 
    
     wherein 
     S=the surface stroke 
     OT=overtravel due to rod weight and accelerations 
     RS=rod stretch due to the fluid load 
     From the above relationship and from FIG. 2, it is evident that the weight of the rod string stores up energy in the form of overtravel at the end of the down stroke which increases the pump stroke. The fluid load produces rod stretch during the up stroke which reduces the pump stroke. Thus, the elasticity and weight of the rod string, as determined by the relative proportions of fiberglass and steel rods, as well as by the elasticity of the rods, particularly of the fiberglass rods, must be tailored to maximize the difference between the overtravel term OT and the rod stretch term RS in the above equation (1). 
     In order to derive the parameters of an optimum rod string, one begins with the basic stress-strain equation: 
     
         ε=σ/E                                        (2) 
    
     where 
     ε=strain 
     σ=stress 
     E=the modulus of elasticity 
     In a steel and fiberglass rod string, the average stress on the fiberglass string, σ 1 , becomes (ignoring the effect of buoyancy on the weight of the rod): ##EQU1## Similarly, the average stress on the steel portion of the rod, .sub.σ s, becomes: ##EQU2## Thus, the total strain on the rod string can be calculated by summing the individual strains of the steel and fiberglass portions of the rod string so that the total strain becomes: ##EQU3## where E f  and E s  represent the modulus of elasticity of fiberglass and steel, respectively; L f  represents the total length of the top (fiberglass) rods in feet; L s  represents the total length of the bottom (steel) rods in feet; E f  represents the modulus of elasticity of the top (fiberglass) rods in pounds per square inch; E s  represents the modulus of elasticity of the bottom (steel) rods in pounds per square inch; and 12 converts feet of rod into inches. 
     The above equation represents the strain on a stationary rod string resulting from gravity without taking into account strains produced by acceleration; however, the effects of acceleration can readily be factored in by modifying the above equation (5) as follows: ##EQU4## where a represents the acceleration imparted to the rod string. Since the amount of elongation of the rod string caused by gravity is constant, only the effect of the variable acceleration contributes to overtravel. Thus, equation (6) which includes the acceleration factor, a, fully defines the overtravel on this downstroke. At the beginning of the upstroke, the rod gains further overtravel as a result of the acceleration of the pumping unit upward (FIG. 2, Diagram III). Thus, at the beginning of the upstroke, the total overtravel of the rod string is equal to the overtravel caused by the acceleration imparted to the rod string by the pumping unit and the overtravel already present at the end of the down stroke and defined by equation (6), this giving a total overtravel of: ##EQU5## where A f  represents the area of the top (fiberglass) rods in square inches; A s  represents the area of the bottom (steel) rods in square inches; W f  represents the weight of the top (fiberglass) rods in air in pounds; and W s  represents the weight of the bottom (steel) rods in air in pounds. 
     The effects of acceleration and gravity on the fluid must also be determined in determining rod stretch. Consequently, the fluid load which is present only on the upstroke, must be calculated. The fluid load is equal to: 
     
         F.sub.66 =(1+a) F                                          (9) 
    
     where F 66  represents accelerated fluid load and F represents fluid load at rest (gravity only). Rod stretch due to this accelerated fluid load is equal to: ##EQU6## where, in equation (10) ##EQU7## Substituting equations (8) and (13) into equation (1): ##EQU8## Rearranging terms and simplifying, the equation for determining the pump stroke becomes: ##EQU9## where S represents the surface stroke in inches; S p  represents the net pump stroke in inches; a represents the acceleration factor which equals Mills&#39; acceleration factor or SN 2  /K, where K equals 70,500 for most pumping units; N represents the number of strokes per minute provided by the surface pumping unit; and F represents the fluid load, which equals 0.34D 2  HG in pounds where D equals the pump diameter in inches, H equals the fluid level in feet and G equals the specific gravity of the fluid. 
     The above equation has been derived for a composite string utilizing two different types of rods, one fiberglass and one steel; however, the equation may be readily generalized for rod strings containing various types of rods having more than two or more diameters and moduli as follows: ##EQU10## wherein the variables are as defined above, with the subscript i indicating variables pertaining to the top rods in the string, the subscript i+1 indicating the rods positioned second from the top of the string, and so on, with the subscript i+j indicating the bottom rods of the string. 
     The above equations have many degrees of freedom and thus permit great flexibility in the design of wells and sucker rod strings. For a new well, it is possible to select a pump diameter and a surface stroke, as well as the make up of the sucker rod string, to achieve optimum pumping capability. However, even for existing wells where the pump diameter and the surface stroke are generally fixed, the make up of the sucker rod string can be optimized for the particular pump and surface stroke used, and it has been found that in practice, substantial improvements in production can be made in existing wells while simultaneously reducing the load on the surface pumping unit and the power required to drive the pumping unit. The improvement in production can be dramatic, and it is often possible to more than double the production of a well by installing a composite string instead of an all-steel string. Moreover, the improvement is achieved without increasing the size of the pumping unit and even with the increased production, the load on the pumping unit is substantially reduced thus making it possible to use a pumping unit that would be too small for use with a conventional steel rod string. Thus, the use of a composite sucker rod string represents a very economical way of substantially increasing production. 
     For example, in a particular 8,000 foot well, by replacing an all-steel sucker rod string with a composite string having the top 5,600 feet fabricated from 0.845 inch diameter fiberglass sucker rods, and the bottom 2,400 feet fabricated from 7/8 inch steel sucker rods, and by utilizing a 11/2 inch diameter pump plunger with a surface stroke of 120 inches at 14 strokes per minute, the production of the well was increased from 200 barrels of fluid per day to 400 barrels of fluid per day utilizing the same surface pumping unit. Moreover, in spite of the increased production, and an increase in pumping speed from a previous 10 strokes per minute, the peak load of the pumping unit was reduced by 27%. Also, the surface stroke of 120 inches resulted in a pump stroke of approximately 150 inches when the composite string was installed. Thus, the composite string provided a 25% increase in pump stroke over the surface stroke. In a typical composite string built in accordance with the principles of the invention utilizing 0.845 inch diameter fiberglass rods, having a modulus of elasticity of approximately 7.2×10 6  lbs/in 2 , which is approximately one-fourth that of steel, the percentage of fiberglass rods is on the order of approximately 50% by length; however, this percentage can vary considerably depending on the diameters of the sucker rods used. When steel sucker rods having a range of diameters, for example, 3/4 inch to 1 inch are used, the percentage of fiberglass rods by length used in a composite string can vary from 20% to 80%. This compares to a typical fiberglass length of 88% to 92% when 11/2 inch steel sinker bars are used. 
     The equations for determining the make up of the composite sucker rod string described in the foregoing are idealized equations, and do not take into account the effects of friction which can result from crooked wells and other sources. Despite the lack of a friction term, the equations have been found to be better than 90% accurate in predicting actual performance. However, the equations can be made even more accurate by adding a constant to the equation defining the pump stroke. Such a constant can be determined empirically by comparing the actual production of a producing well with the predicted production and setting the difference equal to the constant. The effects of friction can also be taken into account by adjusting the weight of the string or the weight of the fluid in the pump stroke equation. 
     Obviously, many modifications and variations of the present invention are possible in light of the above teachings. Thus, it is to be understood that, within the scope of the appended claims, the invention may be practiced otherwise than as specifically described above.

Summary:
A composite sucker rod string for connecting a subsurface well pump to a surface pumping unit utilizes a plurality of relatively elastic sucker rods, such as, polyfilament reinforced, resin bonded sucker rods in conjunction with a plurality of relatively inelastic sucker rods, such as, steel sucker rods, with the lengths and other parameters of the relatively elastic and relatively inelastic portions of the sucker rod string being selected to provide optimum pumping performance within the constraints imposed by the characteristics of the well and pumping equipment.