US 3006541 A Description (OCR text may contain errors) Oct. 31', 1961 F. w. BUBB 3,006,541 PUMPING SYSTEM SIMULATOR Filed March 14, 1958 4 Sheets-Sheet 1 as b m I (I PRESSURE us 1 TRANSDUCER I 1 l I23; TACHOMETER INVENTOR. .w. f F BUBB I2] BY FIG 6 L SERVO I H 3 MOTOR MIXER A T TORNEKS" Oct. 31, 1961 F. w. BUBB 3,006,541 PUMPING SYSTEM SIMULATOR Filed March 14. 1958 4 Sheets-Sheet 2 I I6 V I2 P Pp A K l7 5 A pp NB NB INVENTOR. F.W.BUBB F/G. 5 BY A 7' TORNEVS Oct. 31, 1961 F. w. BUBB 3,006,541 PUMPING SYSTEM SIMULATOR Filed March 14, 1958 4 Sheets-Sheet s 6 c L 0! m C O m m t l I I INVENTOR. F.W.BUBB A 7'TORNEKS Oct. 31, 1961 F. w. BUBB PUMPING SYSTEM SIMULATOR 4 Sheets-Sheet 4 Filed March 14, 1958 mmUDomzxxmlr mmDwwmmE INVENTOR. FW BUBB KOPOE O mmw EMPMEOIUXEI A 7'TO PNEYS United States Patent 3,006,541 PUMPING SYSTEM SIMULATOR Frank W. Bubb, Webster Groves, Mo., assignor to Phillips Petroleum Company, a corporation of Delaware Filed Mar. 14, 1958, $81. No. 721,568 6 Claims. (Cl. 235-61) This invention relates to a simulator which is an analogue of a deep well pumping system. Approximately 85 to 90 percent of present commercial oil producing wells utilize sucker rod pumping systems to move oil from subterranean formations to the surface of the earth. A sucker rod pumping system consists of three principal parts: an engine, a pumping string and a pump. The engine is located at the surface of the earth and is essentially a four-bar linkage which is driven through belts and a gear or chain reducer by an internal combustion engine or an electric motor. This linkage imparts vertical reciprocating motion to a sucker rod string which extends downwardly through the well to the subterranean pump. The pump ordinarily employs two valves. When the plunger is lifted, the traveling valve in the plunger closes and the standing valve at the bottom of the pump barrel opens, the plunger then lifts fluid above the traveling valve, and more fluid flows into the pump barrel through the standing valve. On the downstroke, the traveling valve opens and the standing valve closes, thereby trapping the fluid which was drawn into the barrel on the upstroke so that this trapped fluid flows into the oil column through the travel-ing valve. The oil column surrounds the sucker rod and is contained within the tubing which serves as a conduit for the flow of oil upwardly to the surface. , As the pumping operation is carried out, a column of oil rises in the region between the tubing and the sucker r'od string and thereby produces a viscous drag upon both the tubing and the sucker rod string. The weight and elasticity of the sucker rod string, the tubing and the oil column produce elastic strains in the pumping unit so that the sucker rod string and tubing behave in a manner somewhat analogous to elongated springs. This elastic movement of the sucker rod string and tubing often results in failure of one of the sucker rods, or at least in inefficient operation of the pumping system. It is therefore important that the pumping system be designed and operated in as eflicient a manner as possible. Since it is impossible to observe the actual operation of the down hole portion of a sucker rod pumping system, attempts have been made to predict the operation of such a unit by means of models and simulators. In accordance with the present invention, there is provided an improved simulator for such a deep Well pumping unit. The engine at the surface is simulated by a scale model of an actual well pumping engine. The sucker rod string, oil column and tubing are simulated by means of individual elastic shafts having spaced discs mounted thereon. Rotary motions are imparted to these shafts to simulate vertical movement of corresponding elements in the actual pumping system. The discs are coupled to one another to simulate the viscous drag exerted by the oil column and on the sucker rod string and the tubing. The torques in the shaft segments connecting adjacentdiscs represent the stresses in corresponding segments of the actual pump unit, and the torsional waves running along the shafts correspond to longitudinal waves running along the lines of the actual pumping unit. The downhole pump is simulated by a model of the actual downhole pump. This model actually forces a hydraulic fluid from a tank in a manner which corresponds to the pumping of oil from a reservoir upwardly through the 3,006,541 Patented Oct. 31, 1961 tubing. Forces are applied to the model to simulate the forces which exist in the actual downhole pump unit. Accordingly, it is an object of this invention to provide a system which is an analogue of an actual well pumping unit, which system can be utilized to predict the effect of changing various operating conditions of the actual pumping unit. Another object is to provide a mechanical, hydraulic and electrical system which simulates the operation of a downhole well pumping unit. Other objects, advantages and features of the invention should become apparent from the following detailed description which is taken in conjunction with the accompanying drawing in which: FIGURE 1 is a schematic representation of a typical deep well sucker rod pumping system. FIGURE 2 is a diagram showing forces exerted by the engine at the surface of the pumping unit. FIGURE 3 is a schematic representation of the apparatus of this invention which simulates the operation of the sucker rod string, oil column and tubing of the pumping unit of FIGURE 1. FIGURE 4 is a schematic representation of the apparatus of this invention which simulates the operation of the downhole pump of the pumping system of FIG- URE 1. FIGURE 5 is a diagram showing forces operating on the downhole pump. FlGURE 6 is a second embodiment of a part of the pump simulator. Referring now to the drawing in detail, and to FIG- URE 1 in particular, there is shown a typical deep Well pumping unit which comprises a driving mechanism it) located at the surface of the earth and a downhole pumping assembly 11. Unit 11 includes a well casing 12 which is perforated adjacent an oil producing formation 13. A well tubing 14 extends through casing 12 to a region adjacent formation 13. The downhole pump is carried by the lower end of tubing 14. This pump comprises a barrel 15 which has an opening 16 at the bottom thereof. A standing valve 17 is located adjacent opening 16. A plunger assembly 18 moves through barrel 15 in response to vertical movement of a sucker rod string 2d. Plunger 18 carries a traveling valve 21.. The sucker rod string is attached at its upper end to a polished rod 22 which extends through a stufling box 23 in the top of casing 14. A conduit 24 communicates with casing 14 to remove the pumped oil. Polished rod 22 is attached by an arc type hanger 25 to the first end of a Walking beam 26 which is pivot-- ally mounted on a support 27. The second end of walking beam 26 is connected by a pitman 28 to a crank 29. Crank 2 is connected by a gear or chain reducing assenibly 3%) to a prime mover 31 which can be either an internal combustion engine or an electric motor. A counterbalance weight 32 is mounted on the end of crank 29, and a counterbalance weight 33 can be mounted on the end of walking beam 26 opposite hanger 25. Rotation of crank 29 thus imparts reciprocating vertical movement to the sucker rod string which moves plunger 18 upwardly and downwardly in barrel 15 to force oil to the surface. In accordance with this invention, apparatus is provided which simulates the operation of the actual pumping unit. In order for the two systems to be analogous of one a11- other, it is necessary that the equations describing the behavior of the analogue system be similar to the equations describing the behavior of the actual pumping unit. Thereupon, by suitably adjusting the coefficients of various terms of the analogue system, as by changing the parameters of various parts of the analogue system, a condition is obtained whereby the displacement forces, displacements, velocities and other parameters of the analogue system correspond to the displacement forces, displacements, velocities and other related parameters of the actual pumping unit. At this time the two systems are analogous of one another and further changes in the parameters of the analogue system produce effects thereon which enable accurate prediction of the effects of corresponding changes in the actual pumping unit. The equation describing the operation of the actual pumping unit and the simulator are herein set forth in order to show the correspondence between the two systems. In the equations which follow, the following terminology is employed. Notation for engine M=motor torque. T T T =forces in belts and pitman, see FIGURE 2. J J J J =moments of inertia of elements of FIG- URE 2. 6 6 fi angular displacements of elements of FIG- URE 2. F F F F =friction coefficients of elements of FIG- URE 2. R R R radii of pulleys or flywheels of elements of FIGURE 2. N =torque due to pitman. tx=angular displacement, see FIGURE 2. u =x when 0 is zero. a, b=walking beam dimensions, see FIGURE 2. P stress in polished rod. s=linear displacement of polished rod. K ,K :elastic constants of belts of elements of FIG- URE 2. L lateral engine dimension perpendicular to plane of motion. Notation for pumping string p =stress or load in nth sucker rod segment. s =cross-section area of sucker rod. e =Youngs modulus for sucker rod. x =downward linear displacement of nth mark on sucker rod. l=length of sucker rod lumped segment. w zweig'nt of sucker rod per unit length. g=acceleration of gravity. k =viscous friction coefficient. q =total stress or load in nth oil column segment. s cross-section area of oil column. e =bulk nodulus for oil. y linear displacement of oil upward from nth station point. u =upward velocity of oil at nth station point. w =weight of oil column per unit length. k =viscous coeflicient for oil-tube contact. v =load in nth tube segment. s =cross-section area of tube. e =Youngs modulus for tube. z =downward linear displacement of nth station point of tube. w =weight of tube per unit length. k =viscous friction constant, tube-casing contact. Notation for pumping string simulator P =torque in nth rod shaft segment. E =torsional rigidity for rod shaft segment. X :angular displacement of nth rod disk. G =shear modulus for rod shaft segment. d =diameter of circular rod shaft segment. a =side of square rod shaft segment. L=length of rod, oil and tube shaft segments. I,-=moment of inertia of rod disk. k =viscous friction constant for rod disk. M =torque on rod disk simulating w l. Q =torque in nth oil shaft segment. E =torsional rigidity of oil shaft segment. Y angular displacement of nth oil disk. V =angular velocity of nth oil disk. G shear modulus for oil shaft segment. a =diameter of circular oil shaft segment. l =side of square oil shaft segment. I =moment of inertia of oil disk. K viscous friction constant for oil disk. M =torque on oil disk simulating. R =torque in nth tube shaft segment. E =torsional rigidity for tube shaft segment. Z =angular displacement of nth tube disk. G =shear modulus for tube shaft segment. l =diameter of circular tube shaft segment. K =side of square tube shaft segment. l =moment of inertia of tube disk. K =viscous friction constant for tube disk. M =torque on tube disk simulating w l. Notation for Jump Similarly by inspection, the equations of motion for the flywheel and crank become: where N is the torque on flywheel 3 due to the pitman. This torque N is defined by: N=R T cos 0!. cos +R T sin a sin 0 The equation of motion of the walking beam is: aT cos a cos 0 aT sin a sin 0 bP=J 6 +E 0 (5) The following equations express obvious kinematic relationships between positions of the crank, pitman and walking beam: and L cos t!oL cos DL=Z sin 0 -12 sin 0 Due to stretching of the chain by which the motor drives the flywheel, its length increases by R 0 R 0 and this requires a tension T given by: L sin oc-L sin ot=R +a-R cos (I -a cos 0 where K is an elastic constant. Similarly, The last two equations take account of the spring action on the driving mechanism. Similar effects for the pitman and elastic deflection of the walking beams can be taken as negligible, although they could likewise be included. No attempt need be made at this point to solve these equations, their real purpose being to provide the forms for the equations which the engine simulator must satisfy. In fact, these non-linear differential equations are very diflicult to solve. One purpose of the simulator is to avoid the need for solving such equations. The pumping string comprises, as shown in FIGURE 1, the sucker rod, the oil column and the tubing. Each of these three parts is an elastic system. Instead of expressing the dynamics of these lines by the customary sec- 0nd order linear partial differential wave equations, they shall be treated as lumped parameter systems. Each line is to be divided into 11 segments of length l. The total mass of each segment is to be regarded as concentrated at its midpoint, and each pair of masses is to be regarded as connected by massless elastic connectors. This lumped system is then an approximation to the continuous line. The dynamical equations for these lumped representations of the three pumping string lines can then be expressed. For purposes of illustration, it is assumed that a set of fixed station points are marked off along the fixed casing of the well so as to divide the whole length into n equal segments of length I. It is also assumed that similar marks are scratched on the sucker rod and the tubing when these elements are in the unstressed state, for example, before installation. When the pumping system is in operation, the scratches on the rod jiggle up and down due to the motion caused by the engine. Furthermore, the lengths of the rod and tube segments vary because of the tension and compression waves running up and down. Let x,, be the downward displacement of the nth scratch on the sucker rod from its equilibrium position opposite the nth scratch on the casing. The change in length of the nth rod segment is then x -x The strain in this segment is (x,,x )/l, and the total stress p in the segment becomes: If V is the upward velocity of the oil column at the nth segment, the downward velocity of the nth segment relative to the oil is x +v The viscous drag of the oil on the rod is assumed to be proportional to this relative velocity. For this upward viscous force, it is possible to write k l(x =v Consider now the motion of the segment of rod straddling the point X and extending distances 1/ 2 above and below x This segment has a tension p on its top end, a tension p on its bottom end, a weight w l, and the upward viscous drag k l(a'2 +v Its equation of motion becomes: Equations and 11 are essential dynamical equations for the sucker rod. It is to be understood that these equations apply to every one of the segments into which the rod is regarded as lumped, so that in effect each equation represents a set of n similar equations, distinguished from each other by subscripts. The limiting form of the dynamics expressed above is a single second order linear partial differential wave equation. It is of interest to note how the above equations pass over to the partial ditferential wave equation. Thus, by eliminating p and p (using 10 twice) from (11), there is obtained: r r( x+1 n+ n1) r n r( n+ n) r which, upon passing to the limit by setting 1 0, becomes 6x raT n r 6 Representing the oil velocity by d n yn and integrating, there is obtained: qn o o(yn yn-1) which has the same form as the similar Equation 10 for the sucker rod. The equation of motion of the n-th segment of the oil column must take into account the following forces: the downward viscous drag due to the sucker rod, k l(d- +v the downward viscous drag due to the tube, k l(v +e" where a is the downward velocity of the nth station point of the tube; and the Weight of the oil per unit length, w giving a total weight w l for the segment. Accordingly, the equation of motion of the oil segment is: qn+1 n o n 'h o r 'n+ n) o n+ n) which is similar to the rod Equation 11. Equations 12 and 13 do not completely characterize the behavior of the oil column, as do 10 and 11 for the rod. The oil, being a liquid, cannot support any tension. This means that q cannot be negative, and this limitations on q is expressed by writing: This restriction destroys the continuity of q in 12 and 13, and hence in the partial differential limiting form of these equations. Whatever mathematical difficulties this might cause, it is possible, as pointed out hereinafter, to impose this limitation on the simulator. Actually, the oil is, for the most of the column, under a high compressive stress, so that any tendency for a tension wave to form would amount simply to a decrease in the pressure. Hence, the restriction 14 is of little practical importance. The dynamics of the tubing, except for a change of notation, is the same as that of the rod. Accordingly, it is possible to write down directly the two tubing equations corresponding to Equations 10 and 11, where it is to be understood that the term k lz' is included only in the equations for those segments of tube which actually have oil outside the tubing. It can thus be seen that the dynamics of each one of the three elastic lines constituting the pumping string can be expressed by two equations. The simulator for the pumping string must necessarily be governed by equations of these same forms. From an inspection of the downhole end of FIGURE 1, it can be seen that the pump comprises two parts: the barrel and the plunger, with traveling valve in the plunger and standing valve in the barrel. Freebody diagrams of the barrel and plunger, showing the forces acting on each part, are shown in FIGURE 5. The forces on the plunger are as follows: its weight W is a downward force; the tension p in the bottom segment of the sucker rod is an upward force; the bottom segment of the oil column presses down on the top area of the plunger with a force q s /s where .s is the exposed top area and q /s is the pressure per unit area; and the oil under the plunger and in the barrel produces an upward force p Accordingly, the equation of motion of the plunger is: space above the plunger and the space below, so that the pressure intensity producing the total force p is practically equal to the pressure intensity above the plunger. If the traveling valve is closed, the pressure intensity below the plunger is probably quickly reduced, While the pressure intensity on the ball valve is then equal to the pressure intensity above the plunger, the total force p being intended to represent this unbalanced force at all times. Other discontinuous changes in p occur depending on whether the standing valve is open or closed. It is not necessary to make a detailed study of the way in which p varies because it is possible to simulate the pump by a linear replica which behaves in the same way. Hence, if p is considered to represent this complicated varying force, all that need be done here is to recognize its existence. The freebody diagram in FIGURE shows the forces acting on the barrel. Its weight W acts down; the pressure in the bottom segment of the oil column produces a downward force q s /s on the top annular surface of the barrel flange; a total upward force p is produced on the bottom face of the barrel by the head of oil in the annular space between tube and casing, this force being practically constant; and the projected area of the plunger on the upper face of the bottom of the barrel receives a total pressure p downward. Accordingly, the equation of motion of the barrel is: The force p varies in a complicated discontinuous manner in much the same manner as the force p on the plunger. It is to be understood that this item p includes the small variations in pressure which occur when the ball valve seats and unseats. And again, as with the force p on the plunger, all that need be done here is to recognize the existence of p because the pump simulator behaves in the same manner. The material balance equation for the flow of oil upward through the pump takes the linear form: where the C coeflicients are constants which change discontinuously as the pump passes from one phase of its action to another. An example will clarify the meaning of this equation. Consider that phase of the pump action when the plunger is moving down at velocity ci' relative to the fixed casing; the barrel is moving down at velocity z when the traveling valve is open; the standing valve is closed; and the sealed volume of the barrel below the plunger is filled with oil. Due to the fact that the bottom circular area of the plunger exceeds the annular area of the oil column by the amount s a volume s al' At of oil passes upward through the traveling valve during a time At and joins the oil column. However, due to the downward motion of the barrel, an additional volume s e t which has to be filled appears at the bottom of the oil column. The excess of oil of volume s ai At over the extra volume s z' At flows on up and occupies a volume s v At of the oil column. Accordingly, Cancelling At, there is obtained the material balance equation: for this particular phase of the pump action. In this example the C =s C =s, and C =s A study of all possible phases of the pump action shows that an equation of the form (19) holds, provided the C coefficients are allowed to vary discontinuously from one constant value to another. Detailed analyses of all phases of the pump action are not necessary, because the pump simulator is a linear replica of the actual pump and has to behave in the same manner. The relationships between the equations for the pump and the equations for the simulator are discussed hereinafter, as well as the design details of a rack 8 and pinion connection between the pump simulator and the down hole end of each line of the pumping string. The end conditions which have to be imposed upon the pumping string can now be stated. Since each of the three lines has two ends, there are six end conditions. At top hole, the three end conditions are as follows. For the sucker rod, x =s(:polished rod displacement) (20) the oil produced at the top end of the oil column flows into a storage tank against a back pressure q hence, the pressure in the top segment of the oil column must be equal to this back pressure. Thus, q =q (=storage tank back pressure) (21) The upper end of the tube is always fixed; hence, The three bottom hole end conditions have already been taken into consideration. In writing the dynamical equations for the pump, the bottom hole forces p q and r of the pumping string, as well as the corresponding displacements x y and z were included in the pump Equations 17, 18 and 19. Hence, these three equations provide the needed three down hole end conditions, The simulator for assembly 10 of the actual pumping unit is a scale model of this assembly which is proportioned so as to satisfy all of the necessary equations for both kinematic and dynamic simulation. This model is illustrated in FIGURE 1 and in practice is a small working model of such a unit. The three elastic lines of the pumping string, which comprise the sucker rod, the oil column and the tubing, are simulated by respective flexible shafts 40, 41 and 42 0f FIGURE 3. Shaft 40 carries a plurality of discs 40a, 40b 4011; shaft 41 carries a plurality of discs 41a, 41b Mn; and shaft 42 carries a plurality of discs 42a, 42b 4221. These loading discs thus provide lumped parameters which correspond to the lumped parameter analysis previously made for the actual pumping string. The rotary motion of any one of these discs corresponds to the linear motion of its corresponding actual line segment; the torque in a shaft segment connecting adjacent discs represents the total stress in the corresponding segment of the actual line; the moment of inertia of the discs simulates the mass of the corresponding line segment; and the torsional waves running along the shafts corresponds to the longitudinal waves running along the lines of the actual pumping string. Shaft 40 carries a pinion 45 at one end which meshes with a rack 46. Rack 46 is connected to the polished rod 22 of the pump model so that rotary motion is imparted to pinion 45 corresponding to the vertical motion of polished rod 22. In order to Show that the unit of FIGURE 3 simulates the operation of the pumping string, the equations which describe the operation of the unit of the FIGURE 3 will now be derived. X is employed to designate the angular displacement of the nth rod d disk from its equilibrium position. The angle of twist of the nth rod shaft segment is then X X The torque P in the nth rod shaft segment, being proportional to the angle of twist, can be written as where the constant of proportionality E is given for a shaft segment of circular cross-section of diameter d by the expression: L is the length of the segment of shaft. For a shaft of square cross-section a on the side, E is: 9 see Elements of Strength of Materials by Timoshenko and MacCollough, D. Van Nostrand Company, Inc., page 266. It should be evident, however, that other elastic elements can be employed. Reference is now made to the Equation 11 of motion for the nth segment of the actual sucker rod. The terms in Equation 11 are considered separately. The nth rod segment of the similar exerts on the nth disk the torque P as seen from the previous discussion. The n-l-lst rod segment exerts the torque P The unbalanced torque on the nth rod disk is then P P which corresponds to the left side of Equation 11. in order to simulate the dead weight w l of the nth segment of the actual sucker rod, there is exerted on the nth disk a corresponding torque M This is accomplished by connecting a motor 47n to disk 40n through a connecting disk 48n. The torque provided by this motor can be varied by controlling the current supplied to the motor, thus providing for a change of torque when a new problem is set up on the simulator. The viscous resistance term k l(r'v +v due to the drag of the oil column on the sucker rod must be represented in the equation of motion for the nth rod disk. This term in the simulator equation is provided by a magnetic clutch 50n having one plate connected to the nth rod disk 4012 through a disk SM and having the other plate connected to the nth oil column disk 41n through a disk 52m. The two clutch plates rotate at a relative velocity X -l-V which is proportional to the relative velocity al= +v of the actual system. The coupling between the clutch plates can be adjusted so that the viscous torque between the plates is proportional to the relative velocity. This provides a torque K (X +V which acts in proper direction on both the rod disk and the column disk. The magnitude of this torque can be controlled by varying the current producing the magnetic field between the plates. And finally, the kinetic reaction w lx /g of (11) is represented in the simulator by the term I X /g, where I is the moment of inertia of the rod disk and X is its angular acceleration. Accordingly, the equation of motion for the nth rod disk is: which is of the same form as Equation 11 for the actual sucker rod. X is employed to designate the angular displacement of the nth oil disk from its equilibrium position. The angle of twist of the nth oil shaft segment is then Y and the torque in this oil shaft segment is: which has the same form as the corresponding Equation 13 for the actual oil column segment. The dead weight terms are represented by motors 54 which are coupled to disks er by disks 55. The viscous drag between the oil and tubing columns is simulated by clutches 56 which are connected to disks 41 by disks 57 and to disks 42 by disks .In order to simulate the restriction of Equation 14 which states that the oil column cannot stand any tension, a ratchet 60 is employed at one end of each oil shaft segment where it ties into the oil disk. This device releases the connection between oil shaft segment and oil disk whenever the torque tries to correspond to a tension in the coil column, and takes hold when the torque is in the sense corresponding to compression. Hence, Qn t (14) which corresponds to the inequality of Equation 14 for the actual oil column. The rate of rotation V =Y of the top oil shaft segment is proportional to the production B of the well, so that: cordingly, it is possible to write: n' t( n n-l) and 1.2,. n+1 n 0( n+ n)+ t n t which have the same forms as Equations 15 and 16 for the actual tubing. Motors 70 are connected to disks 42 by disks 71 to represent the dead weight terms. Before discussing the end conditions which must be imposed upon the three shafts simulating the pumping string, it is to be noted that a finite difierence analysis cannot, from its mathematical nature, be accurate except up to one half the element of extension. That is, in the present case, to within the half element of length [/2 for the actual system and L/ 2 for simulator. In order to correct partially for such small errors, it is possible to round out the ends with half loads. In the present instance, this means that the disks on the ends of the shafts can have moments of inertia one-half those of the intermediate disks. These half-disks need not be recognized explicitly in the equations, however. As mentioned previously, a linear pump simulator is employed in the simulator of this invention. The equations for this pump simulator are of exactly the same form as the corresponding Equations 17, 18 and 19 for the actual pump, and for this reason there is no need to set them up again. When applying these equations to the pump simulator, a bar will be employed above the symbols. The end conditions which have to be imposed upon the three lines of the pumping string simulator must correspond properly to the end conditions on the actual pumping String as explained previously. At top hole end of the sucker rod shaft, there is placed the pinion 45 of pitch radius r in mesh with the rack 46 representing the polished rod of the engine simulator. This pinion, upon rotating through the angle X, must produce a linear displacement 5 equal to that of the polished rod, so that rK =5. The pinion must also convert the torque P into the force T in the rack, so that rT=P The top hole end condition on the sucker rod shaft are thus provide if the two equations: rX =x and rT=P are satisfied. This is the first time in writing the equations for the simulator that there have been obtained equations which are'not exactly of the same form as the corresponding equations for the actual system. This is due, of course, to the introduction of the rack and pinion connection, corresponding to which there are no parts in the actual system. However, so far as the use which is to be made of Equation (2 is concerned, it does provide the simulator equivalent of Equation 20. Corresponding to the storage tank back pressure q against which the oil column discharges, there is applied to the top hole end of the oil shaft a torque Q by a motor 62 which is connected to the top hole end of the oil shaft by disks 63 and 63'. Thus, corresponding to Equation 22. The three bottom hole conditions are taken care of for the pumping system simulator as follows. The rotary displacement X of the bottom hole end of the rod shaft is converted into the corresponding linear motion i of the plunger by a pinion 65 of pitch radius r on the bottom hole end of the rod shaft and a rack 66 on the plunger simulator, the relationship being rR =x At the same time, this rack and pinion provide the correct relation P =r5 between the torque P in the bottom segment of the rod shaft and the force p on the plunger simulator. Thus, rX =x and which, taken together, provide the required equivalent of the actual pumping system end condition at the down hole end of the sucker rod. In similar fashion, a rack 67 and pinion 68 provide the equations rZ =E and for realizing the proper equivalent of the bottom hole end condition on the tube. The simulator of the downhole pump is illustrated in FIGURE 4. As previously mentioned, this simulator is essentially a scale model of the actual pump, and elements corresponding to the elements of the pump of FIGURE 1 are indicated by like primed reference numerals. The simulator comprises a tank 12' which represents the oil reservoir contained within casing 12 of FIGURE 1. A cup-shaped member 75 is disposed within container 12' to represent the lower end of tubing 14. The pump assembly is positioned within member 75. A fixed plate 76 extends across the top of member 75, and the interior of member 75 is maintained under pressure by means of a gas which is supplied thereto from a source, not shown, through a conduit 77. A pressure gage 78 and a valve 7 9 are disposed in conduit 77 to permit the pressure within member 75 to be regulated. The pump plunger is attached to a rod which terminates at its upper end in rack 66. A spring 80 extends between plate 76 and a fixed point on rod 20' so as to exert a force representative of the weight of the pump plunger. Springs 81 extend between container 12' and member 75 to exert forces which simulate the weight of the pump barrel. Member 75 is attached to a rod 14 which terminates at its upper end in rack 67. Member 75 is thus free to move in a vertical direction within container 12 on bearings 82 which are held by a rigid support 83. Container 12 and member 75 are filled with a quantity of hydraulic fluid which simulates the oil in the actual pumping system. The vertical reciprocating movement of rod 20' actuates the pump to force this hydraulic fluid from the interior of member 75 through a conduit 85. Conduit communicates at its second end with a re servoir 86. A pump 87 is disposed within conduit 85. A conduit 88, having a pump 89 and a valve 90 therein, communicates between reservoir 86 and container 12. The fluid pumped from member 75 is thus recycled to container 12 to supply a new quantity of oil to be pumped. Reservoir 86 can also be maintained under a desired pressure by a source of gas, not shown, which communicates with the reservoir through a conduit 91 which has a pressure regulator 92 and a valve 93 therein. Pump 89 is actuated by a pressure transducer 94 which measures the pressure within container 12. Pump 89 is thus regulated so as to maintain a desired operating pressure within container 12 which simulates the reservoir pressure. A flow transducer senses the rate of fluid flow through conduit 85 which represents the rate at which oil is pumped from the reservoir. This transducer provides an output signal which is applied through a mixer 101 and an amplifier 102 to actuate a servo motor 103. The drive shaft of motor 103 is connected through a tachometer-generator 104 to a gear 106 which meshes with a gear 107. Gear 107 is connected to shaft 41 through a ratchet 108. Shaft 41 is thus rotated in a manner representative of the flow of fluid through conduit 85. Tachometer-generator 104 provides an output signal which is applied as a feedback signal to the input of mixer 101. A torque transducer provides an output signal representative of the torque applied to shaft 41. This can be measured by a strain gage 109 on shaft 41. The output signal from transducer 105 is applied through a mixer 110 and an amplifier 111 to actuate a second servo motor 112. The drive shaft of motor 112 is connected to pump 87 in conduit 85. A pressure transducer 113 senses the pressure in conduit 85 and provides an output feed-back signal to the input of mixer 110. This bottom hole servo system is based upon the principle that any force applied to the plunger rod of the simulator pump which is not balanced against the hydraulic pressures in the system produces a flow which is measured by transducer 100. Any hydraulic pressure which is not balanced by a force on the plunger of the simulator pump produces a motion of the plunger. The apparatus is designed and operated so that a flow in the hydraulic system through conduit 85 is made to produce a rotation of the oil column simulator at a rate which is directly proportional to the flow in the hydraulic system. The torque at the input of the oil column simulator is made to produce a pressure through pump 87 in the hydraulic system which is directly proportional to the torque. Under these condtions, the plunger is loaded in the same manner as if it were driving the oil column simulator directly. The various circuit elements shown in FIGURE 4 can be convention apparatus providing electric signals representative of the quantities being measured. A simplified embodiment of the pump simulator is illustrated in FIGURE 6. Motor 122 is geared directly to gear 107 on shaft 41, as is a tachometer-generator 120. The output signal of tachometer is compared with the output signal of pressure transducer 113 by a mixer 121. Any difference between the signals being compared actuates a servo motor 122 to increase the rate of rotation of shaft 41 and the speed of pump 87. The rotation of shaft 41 is thus an accurate representation of the rate at which oil is lifted by the pump unit. The simulator of this invention can thus be employed to study the operation of an actual pumping assembly. The parts of the simulator are constructed to be representative of the corresponding parts of the actual assembly. Stresses in the actual unit can be measured by measuring the torque in the corresponding part of the simulator, as by the use of a strain gage. Tachometer 61 measures the rate of oil production at the surface. The 13 effect of changes in pumping speeds and applied forces can readily be observed by use of this simulator. While the invention has been described in conjunction with present preferred embodiments, it obviously is not limited thereto. 7 What is claimed is: I 1. Apparatus for simulating a pumping system which includes a sucker rod string, tubing surrounding said sucker rod string, an oil column between said tubing and said string, a first downhole pump actuated by said string to elevate said oil column, and means positioned at the surface to raise and lower said string, comprising: a plurality of units connected together, each including a first shaft representing a tubing segment, a second shaft representing an oil column segment, a third shaft representing a string segment, a disk coupled to each shaft and having a moment of inertia representing the inertia of the corresponding segment, a drag means coupled to each shaft to exert a force thereon retarding rotary motion in a manner representative of the weight of the corresponding segment, means coupling the oil shaft and the tubing shaft to simulate viscous drag between the oil column segment and the tubing segment, means coupling the oil column shaft and the string shaft to simulate the viscous drag between the oil column segment and the string segment, and clutch means in the oil columnshaft to permit rotation in only one direction; means coupled to the first end of said string shaft to impart rotary motion thereto to simulate said means positioned at the surface; means retaining the corresponding first end of said tubing shaft stationary; a drag means coupled to the corresponding first end of said oil column shaft to simulate pressure tending to prevent oil from being discharged at the surface; a tank having hydraulic fluid disposed therein; a second downhole pump in said tank, said second pump being of substantially the same configuration as said first pump; means connecting the second end of said string shaft to said second pump to actuate said second pump in response to rotary motion of said string shaft; a conduit communicating with said tank to receive the flow of fluid therefrom, a third pump in said conduit, means coupling said third pump to said oil column shaft; means to establish a first signal representative of the rate of rotation of said oil column shaft; means to establish a second signal representative of the pressure in said conduit; means to compare said first and second signals; and means responsive to said means to compare to vary the rate of rotation of said oil column shaft and said third pump. 2. Apparatus for simulating a pumping system which includes a sucker rod string, tubing surrounding said sucker rod string, an oil column between said tubing and said string, a first downhole pump actuated by said string to elevate said oil column, and means positioned at the surface to raise and lower said string, comprising: a plurality of units connected together, each including a first shaft representing a tubing segment, a second shaft representing an oil column segment, a third shaft representing a string segment, a disk coupled to each shaft and having a moment of inertia representing the inertia of the corresponding segment, a drag means coupled to each shaft to exert a force thereon retarding rotary motion in a manner representative of the weight of the corresponding segment, means coupling the oil shaft and the tubing shaft to simulate viscous drag between the oil column segment and the tubing segment, means coupling the oil column shaft and the string shaft to simulate the viscous drag between the oil column segment and the string segment, and clutch means in the oil column shaft to permit rotation in only one direction; means coupled to the first end of said string shaft to impart rotary motion thereto to simulate said means positioned at the surface; means retaining the corresponding first end of said tubing shaft stationary; a drag means coupled to the corresponding first end of said oil column shaft to simulate pressure tending to prevent oil from being discharged at the surface; a tank having hydraulicfluid disposed therein; a second downhole pump in said tank, said second pump being of substantially the same configuration as said first pump; means connecting the second end of said string shaft to said second pump to actuate said second pump in response to rotary motion of said string shaft; a conduit communicating with said tank to receive the flow of fluid therefrom; a third pump in said conduit; means to establish a first signal representative of the rate of flow through said conduit; means to establish a second signal representative of the rate of rotation of said oil column shaft; first means to compare said first and sec ond signals; means responsive to said first means to compare to impart rotary motion to said oil column shaft; means to establish a third signal representative of the pressure in said conduit; means to establish a fourth signal representative of the torque in said oil column shaft; second means to compare said third and fourth signals; and means responsive to said second means to compare to energize said third pump. 3. Apparatus for simulating a pumping system which includes a sucker rod string, tubing surrounding said sucker rod string, an oil column between said tubing and said string, a downhole pump actuated by said string to elevate said oil column, and means positioned at the surface to raise and lower said string, comprising: a plurality of units connected together, each including a first shaft representing a tubing segment, a second shaft representing an oil column segment, a third shaft representing a string segment, a disk coupled to each shaft and having a moment of inertia representing the inertia of the corresponding segment, a drag means coupled to each shaft to exert a force thereon retarding rotary motion in a manner representative of the weight of the corresponding segment, means coupling the oil shaft and the tubing shaft to simulate viscous drag between the oil column segment and the tubing segment, means coupling the oil column shaft and the string shaft to simulate the viscous drag between the oil column segment and the string segment, and clutch means in the oil column shaft to permit rotation in only one direction; means coupled to the first end of said string shaft to impart rotary motion thereto to simulate said means positioned at the surface; means retaining the corresponding first end of said tubing shaft stationary; a drag means coupled to the corresponding first end of said oil column shaft to simulate pressure tending to prevent oil from being discharged at the surface; a first stationary vessel containing hydraulic fluid; a second cup-shaped vessel suspended within said first vessel; means connecting said second vessel to the second end of said tubing shaft so that rotary motion of said tubing shaft results in reciprocating motion of said second vessel within said first vessel; a pump barrel carried by said second vessel; a plunger disposed within said barrel; means connecting said plunger to the second end of said string shaft so that rotary motion of said string shaft results in reciprocating motion of said plunger within said barrel; and means responsive to the flow of fluid from said second vessel to rotate the second end of said oil column shaft at a rate representative of the rate of flow of fluid from said tank. 4. The apparatus of claim 3 further comprising means to maintain the fluid in said first vessel under pressure to simulate the reservoir pressure of the formation being produced. 5. Apparatus to simulate a well pumping unit compris ing first, second and third shafts to simulate the sucker rod string, tubing string and oil column, respectively, of a pumping unit, a plurality of disks mounted in spaced relationship with one another on each of said shafts, a plurality of drag means coupled to each of said shafts to retard rotary movement thereof, a plurality of means coupling said third shaft to said second shaft to transmit rotary motion therebetween, a plurality of means coupling said third shaft to said first shaft to transmit rotary motion therebetween, a plurality of clutch means in said third shaft so that rotary motion is transmitted to adjacent segments only in one direction, means coupled to one end of said first shaft to impart reciprocating rotary motion thereto, means retaining the corresponding first end of said second shaft stationary, drag means coupled to the corresponding first end of said third shaft to tend to prevent rotation thereof, a tank having hydraulic fluid disposed therein, a first pump in said tank, means coupling the second end of said first shaft to said first pump to actuate same responsive to rotary motion of said first shaft, a conduit communicating with said tank to receive the flow of fluid therefrom, a second pump in said conduit, means coupling said second pump to said third shaft, means to establish a first signal representative of the rate of rotation of said third shaft, means to establish a second signal representative of the pressure in said conduit, means to compare said first and second signals, and means responsive to said means to compare to vary the rate of rotation of said third shaft and said second pump. 6. Apparatus to simulate a well pumping unit comprising first, second and third shafts to simulate the sucker rod string, tubing string and oil column, respectively, of a pumping unit, a plurality of disks mounted in spaced relationship with one another on each of said shafts, a plurality of drag means coupled to each of said shafts to retard rotary movement thereof, a plurality of means coupling said third shaft to said second shaft to transmit rotary motion therebetween, a plurality of means con- 16 pling said third shaft to said first shaft to transmit rotary motion therebetween, a plurality of clutch means in said third shaft so that rotary motion is transmitted to adjacent segments only in one direction, means coupled to one end of said first shaft to impart reciprocating rotary motion thereto, means retaining the corresponding first end of said second shaft stationary, drag means coupled to the corresponding first end of said third shaft to tend to prevent rotation thereof, a tank having hydraulic fluid disposed therein, a first pump in said tank, means coupling the second end of said first shaft to said first pump to actuate same responsive to rotary motion of said first shaft, a conduit communicating with said tank to receive the flow of fluid therefrom, a second pump in said conduit, means to establish a first signal representative of the rate of flow through said conduit, means to establish a second signal representative of the rate of rotation of said third shaft, first means to compare said first and second signals, means responsive to said first means to compare to impart rotary motion to said third shaft, means to establish a third signal representative of the pressure in said conduit, means to establish a fourth signal representative of the torque in said third shaft, second means to compare said third and fourth signals, and means responsive to said second means to energize said second pump. References Cited in the file of this patent UNITED STATES PATENTS 2,698,133 Bubb Dec. 28, 1954 Patent Citations
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