Method for reducing sand production in submersible-pump wells
BACKGROUND 
     The Appendix to this disclosure contains copyrighted material. Permission is granted to make copies of the Appendix solely in connection with making copies of the patent, and for no other purpose. 
     This application relates to well production and electric submersible pump (ESP) systems; specifically, it relates to a method for aiding selection and optimization of pump apparatus by simulating the well drawdown process. 
     Basics of Pumping Apparatus 
     FIG. 1 is a simplified illustration of a wellbore configuration. The well comprises a casing 10 which extends downward from the surface to a formation 12 which contains a fluid under pressure. Casing 10 has numerous perforations 14 in the region adjoining formation 12 to allow fluid to pass from the formation 12 into the casing 10. The region of the casing having perforations is sometimes referred to as the &#34;perforation zone&#34;. 
     Inside casing 10 is a production tubing assembly 16 which hangs vertically within casing 10 from the aboveground well structure (not depicted). Production tubing assembly 16 comprises a pump 18 having an intake for the well fluid, and usually also includes a number of components not discussed here such as motors, electrical cables, and gas separators. 
     The intake for pump 18 is submerged in the well fluid, which partially fills casing 10. Perforations 14 allow well fluid to pass into casing 10 and partially fill it, to a level 20. Fluid level 20 is referred to as the &#34;static fluid level&#34;, because it is the level to which the column of fluid rises when the pump is inactive. 
     Fluid is forced to the static level 20 by the upward force of the formation pressure, which is designated SIBHP (shut-in bottom hole pressure) for an inactive, or &#34;shut in&#34;, well. When the downward weight of the column of fluid equals the force of the fluid pushed upward by the formation pressure, fluid stops rising within the casing. 
     Well Drawdown 
     Well drawdown refers to the process occurring upon startup of pump 18. The pump activity reduces fluid pressure in the area surrounding the pump intake. This causes the level of fluid within casing 10 to go down; as the pressure of formation 12 is released, the weight of the casing column of fluid will oppose this pressure reduction. The column of fluid will drop from its static level 20 until equilibrium is reached again at a producing level 22, as depicted in FIG. 2. 
     FIG. 2 depicts the actions taking place upon startup of the pump 18 of FIG. 1 Pump 18 extracts fluid at a rate Q t  consisting of a portion of formation fluid Q f  and a portion of fluid Q a  from the casing-tubing annulus: 
     
         Q.sub.t =Q.sub.a +Q.sub.f 
    
     Fluid level 22 is the level reached when the pump is operational and the upward force of the formation pressure balances the downward force of the column of fluid. When level 22 is reached, the well is said to be stabilized and producing. 
     The drop from level 20 to level 22 corresponds to a pressure change ΔP in the pump region of the formation, so that the pressure P wf  in the pump region is reduced from its original value by an amount ΔP: 
     
         P.sub.wf =SIBHP-ΔP 
    
     When the well has reached its stabilized producing stage, all of the fluid pumped then comes from the formation (i.e., Q t  =Q f  and Q a  =0). 
     The Vogel Relation 
     The relationship between production rates and formation pressures has been modelled in several approximate relationships. One widely used approximation is the Vogel relation: 
     
         Q.sub.o /Q.sub.o.sbsb.max =1-0.2(P.sub.wf /P.sub.r)-0.8(P.sub.wf /P.sub.r).sup.2 
    
     Using this relation, one can find Q o .sbsb.max, or the maximum rate of production. P r  in this equation is static pressure (SIBHP), and P wf  is pressure of the formation. FIG. 4 is a graph of the Vogel relation. 
     Vogel&#39;s relation is not the only approximation used to aid in predicting well performance. For example, another well-known approximation is the straight-line productivity index: 
     
         Q.sub.o =Q.sub.o.sbsb.max (1-P.sub.wf /P.sub.r) 
    
     The straight-line productivity index may be used in similar fashion to the Vogel relation above to determine the maximum rate of production. 
     Dynamic Head 
     When selecting a submersible pump for a given well, it is necessary to select one powerful enough to overcome the difference in elevation between the fluid column in the casing and the surface. This difference is often called the &#34;head&#34; or &#34;dynamic head&#34;. Dynamic head also may refer to the pressure in the wellbore as a result of this vertical difference. 
     In electric submersible pump systems, the total dynamic head (TDH) is commonly taken as the sum of vertical lift, frictional loss, and any surface pressure: 
     
         TDH=H.sub.lift +H.sub.friction 30 H.sub.surface pressure 
    
     The vertical lift component is the primary component of TDH, corresponding to the difference in elevation. Friction loss may be calculated for a given length and diameter of tubing using methods well known to those of ordinary skill. Surface pressure is any back pressure or pressure in tubing at the surface impeding the production of fluid. 
     Sand in Formations 
     One major problem in submersible pump system has been that the well drawdown process typically causes the movement of sand particles with a formation. If the formation pressure P wf  changes abruptly, as might occur upon startup of the well pump, pressure differences within the formation may loosen or wash away sand particles, and cause sane to be produced with the well fluid. 
     Production of sand is highly undesirable. Sand passing through the pump and intake causes premature wear and abrasion, and shortens the useful life of the pump. 
     Referring to FIGS. 3a and 3b, a cross-section of wellbore 10 occupies only a very small area of a typical formation 12. In FIG. 3a, the well is assumed to be shut in, and P wf  =P r , the shut in bottom hole pressure. The pressure throughout the formation is P r  in all locations. FIG. 3b shows the formation immediately after drawdown begins. At the wellbore 10, formation pressure P wf  is reduced. However, at some distance x from the wellbore, the formation pressure is still P r  ; pressure does not instantaneously change for all locations within the formation. 
     The pressure difference P r  -P wf  causes heavy instantaneous fluid flow, increasing the probability that the fluid carries sand with it. Thus if the rate of change of pressure dP wf  /dt is lessened, sand production is inhibited. 
     SUMMARY OF THE INVENTION 
     In accordance with the invention, a method for determining a desirable well startup profile minimizes sand production by ensuring that pressure differentials within a formation remain within prescribed limits throughout the drawdown process. 
     An iterative technique may be used to accurately estimate, from fixed pump parameters and well performance data, the rate of change of formation pressure dP wf  /dt. Fixed pump parameters include the particular pump configuration used, the size and length of tubing, and the size of the well casing. 
     To reduce the rate of change of formation pressure, one may limit the initial rate of production, keeping dQ t  /dt low so as to keep dP wf  /dt low. 
     The production rate may be controlled using two techniques or a combination of the two: (1) varying the frequency of AC power delivered to the pump motor (slowing the pump action); or (2) applying back pressure to impede the flow of fluid produced (e.g., controlling production flow using a valve or the like). Implementation of a startup profile according to the invention may be controlled through a computer, monitoring and altering production rates as needed. 
     In fact, drawdown may be controlled using little or no feedback if a desired profile (rate of fluid production versus time) is determined in advance of well startup, and used to direct computer control of drawdown. A computer or terminal unit at the well site may regulate production rates according to a preset profile, enabling more precise and accurate control of well startup. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS 
     The invention, while particularly set forth in the appended claims, may be understood more easily upon reading the following detailed description of specific embodiments, in which: 
     FIG. 1 is a simplified illustration of an inactive (non-pumping) wellbore; 
     FIG. 2 depicts the wellbore of FIG. 1 upon startup of the pump 18; 
     FIGS. 3a and 3b represent a formation taken along the line 1--1 of FIG. 1 and depict a shut in and a producing well, respectively; 
     FIG. 4 is a graph of Vogel&#39;s inflow performance relation; 
     FIG. 5 is a block diagram depicting a control system which varies the frequency of power input to a motor; 
     FIG. 6 is a block diagram depicting a control system which uses an adjustable choke; 
     FIG. 7 is a block diagram depicting a control system which uses a diverter valve to return fluid to the casing; 
     FIG. 8 depicts a rate from formation that has a large initial variation with time; 
     FIG. 9 depicts a rate from formation varying linearly with time; 
     FIG. 10 is a graph depicting a family of curves for TDH at different frequencies, with a production rate superimposed on those curves; 
     FIG. 11 is a graph of power frequency versus time to implement a selected production rate profile from the example. 
    
    
     DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS 
     1. Profile Control Using Input Power Frequency 
     Using a variable-frequency motor in a submersible pump allows a high degree of control over the drawdown process. In brief, when the frequency of AC current is increased or decreased, the pump rate is increased or decreased also in a linear proportion: 
     
         Pump Rate.sub.f =(f/60)Pump Rate.sub.60Hz 
    
     For a frequency f, the pump rate is (f/60) times the 60 Hz pump rate. Likewise, the total dynamic head required is dependent on the power input frequency f squared, so that: 
     
         TDH.sub.f =(f/60).sup.2 TDH.sub.60Hz 
    
     FIG. 5 depicts a control system having a processor 24, variable frequency drive 26, and pump system 28. Feedback from the drive 26 and pump 28 indicate to the processor whether it should increase or decrease the input frequency of the drive. Signals which may be monitored include pressure at the pump intake, surface pressure, fluid level in the casing, and rate of production. 
     2. Profile Control Using Back Pressure 
     FIG. 6 illustrates a control system which limits the rate of production using a variable choke. Feedback signals such as production rate, fluid level, pressure at the pump intake, and choke position may be returned to the processor 24. Processor 24 opens or closes choke 30 by the degree necessary to optimize drawdown. 
     The power input frequency of pump 28 is not altered in this example, having no signal input from the processor 24. However, the pump rate, or rate of fluid through the pump, is varied through surface pressure resulting from choke 30. The necessary surface pressure profile may be predetermined for the processor 24 for a given well and pump configuration to automate the operation of choke 30. 
     3. Profile Control Using Fluid Circulation 
     Control of the drawdown process may also be accomplished through the use of a diverter valve as in FIG. 7. Under control of the processor 24, the diverter 32 recirculates an amount of fluid Q c , sending it back into the casing to pass through pump 28 another time. Total fluid produced Q p  is equal to Q t , the total through the pump, minus Q c  : 
     
         Q.sub.p =Q.sub.t -Q.sub.c 
    
     Feedback of the diverter&#39;s position and the amount of recirculated fluid enable the processor 24 to control the diverter 32 opening much in the same manner that it controls choke 30 in Example 2. 
     4. Control With Advance Selection of Desired Profile 
     The methods described above provide a limited measure of control over the drawdown process. Greater control and optimization of drawdown may be achieved by programming an on-site computer such as processor 24 to implement a predetermined profile of fluid production rate versus time. 
     Determination of an optimal profile can be made using iterative techniques which predict the effects of uncompensated (fixed production rate) drawdown to enable the user to select a preferable drawdown. The following example will help in illustrating this. 
     Example: Pump Selection 
     Given the following hypothetical data, one can select a suitable size pump for a specific well and predict its performance: 
     SIBHP=1700 psi; P wf  =1692 psi; 
     Q 0  =200 BPD in initial test 
     Center of perforations: 5200&#39; 
     Desired production rate Q 0  =1440 BPD 
     Tubing: 2.5&#34; diameter, in average condition 
     Specific gravity of water=1.07 
     Formation pressure P r  =285 psi 
     Using Vogel&#39;s equation described earlier, one can solve for the maximum fluid production rate Q 0 .sbsb.max =23661 BPD. The desired eventual rate is Q 0  =1440 BPD, so Vogel&#39;s equation may be solved again with P wf  as the unknown to determine the formation pressure of the well when producing. After translating the calculated pressure into feet of head, the result is P wf  =3618 ft. above the perforation center. 
     The working fluid level forms the vertical lift component of head; WFL=5200-3618=1582 ft. Surface pressure in feet of head is also calculated: 
     
         H.sub.sp =(285 psi)(2.31 ft/psi)/1.07=628 ft. 
    
     Frictional loss in the tubing may be accounted for using the Hazen-Williams method well known in the art. In this example, the loss amounts to 87 feet in head. 
     Total dynamic head is the sum of these three components: 
     
         TDH=1582+628+87=2298 ft. 
    
     For this hypothetical example, a 132-stage DN1300, manufactured by Reda-Camco, provides adequate pumping capability. 
     Simulation of Drawdown 
     When a pump has been selected to match the well, then a simulation program can be run to determine the effects of uncompensated drawdown, using pump and well test data as initial variables. The Appendix contains BASIC source code for one such program. 
     The simulation program is iterative, calculating a production rate Q t .sbsb.n on each iteration n, based on fluid level and formation pressure P wf .sbsb.n. During the simulated time interval t between iterations n and (n+1), a volume of fluid is produced which may be used to determine a new fluid level and formation pressure P wf .sbsb.n+1. A new pump rate Q t .sbsb.n+1 is then calculated. This iterative technique continues until the pump rate has stabilized. 
     The drop x in fluid level on successive iterations can be used to determine the portion of fluid Q a  from the casing-tubing annulus. The rate from formation Q f  =Q t  -Q a  may thus be determined. 
     A typical uncompensated well will draw down with a rate from formation that varies with time as in FIG. 8. The user may then choose a preferred profile which reduces the slope of this plot as much as possible. FIG. 9 depicts one such design choice. The rate from formation changes at a constant rate. Other profiles may be selected, of course, depending on the user&#39;s specific needs. 
     Simulation of Compensated Drawdown 
     Following selection of a desired formation rate profile, it is necessary to determine a corresponding profile of production rate Q t  versus time. Another iterative test, similar to the simulation program described above, can be used to determine the production rate profile to implement in controls. 
     Controlling the production rate may be accomplished using one of the methods discussed earlier: (1) varying power input frequency to a variable-speed pump; (2) adjusting a choke to limit fluid passage; and (3) diverting fluid back into the well. This example will illustrate the use of a variable-speed pump. 
     Total Dynamic Head 
     Total dynamic head varies with the power input as shown earlier: 
     
         TDH.sub.f =(f/60).sup.2 TDH.sub.60Hz 
    
     The TDH curve may be expanded into a family of curves for different frequencies as shown in FIG. 10. Against this family of curves is plotted the production rates (which change in dynamic head as the surface pressure and fluid level in the casing changes). 
     The TDH curve for a pump is usually available from the pump&#39;s manufacturer, but may also be represented mathematically through regression analysis as a polynomial 
     
         A.sub.n Q.sup.n +A.sub.n-1 Q.sup.n-1 +. . . +A.sub.1 Q+A.sub.0 
    
     For the simulation performed in the course of implementing this invention, the polynomials used were from 8th- to 15th-degree. 
     From the plots of FIG. 10, it is possible to determine a profile of power input fequency versus time corresponding to the earlier profile of fluid production rate versus time. This frequency-time relationship, depicted in FIG. 11, may be implemented using a control system as shown in FIG. 5. 
     It will, of course, be apparent to those of ordinary skill having the benefit of this disclosure that the above embodiments do not represent all of the ways that the invention may be practiced. Thus it is noted that the invention is intended to be limited only by the scope of the appended claims. ##SPC1##