Automatic Iteration of Upstroke and Downstroke Damping Factors with Iterative Bisection Method

Abstract

A method for calculating a downhole card for an oil and gas well includes setting a damping factor to an initial value, calculating a downhole card using the initial load value, and determining whether the difference between a true load value and an effective load value is within a desired error tolerance. If not, the method can also include setting a new damping factor to a value equal at a mid-point between the initial damping factor and a damping factor bound, recalculating the downhole card using the new damping factor, and determining whether the difference between a new true load value and a new effective load value is within a desired error tolerance. If not, the method can also include setting the new damping factor to a value equal at a mid-point between the previous damping factor and a damping factor bound.

Description

BACKGROUND OF THE INVENTION

When a well's own energy is not enough to bring the produced fluids to the surface, an artificial lift system is needed to help lift the fluid from the wellbore. One of these methods is sucker rod pumps. In sucker rod pumps, the circular motion of crank at the surface is translated downhole to the pump using a polished rod and rod string.

Referring now to FIGS. 1, 1A and 1B, a diagram of a typical sucker rod pump used in oil wells is described. The sucker rod pump is described only for the purposes of illustrating the operation of a typical oil well and is not intended to be limiting in any manner as the present invention is applicable to any producing oil well including those using any means of artificial lift, such as rod pumping, electric submersible pumps, progressive cavity, and other methods.

Well 10 includes well bore 11 and pump assembly 12. Pump assembly 12 is formed by a motor 13 that supplies power to a gear box 14. Gear box 14 is operable to reduce the angular velocity produced by motor 13 and to increase the torque relative to the input of motor 13. The input of motor 13 is used to turn crank 15 and lift counter weight 16. As crank 15 is connected to walking beam 17 via pitman arm 18, walking beam 17 pivots and submerges plunger 19 in well bore 11 using bridle 20 connected to walking beam 18 by horse head 21. Walking beam 17 is supported by Sampson post 22.

Well bore 11 includes casing 23 and tubing 24 extending inside casing 23. Sucker rod 25 extends through the interior of tubing 24 to plunger 19. At the bottom 25 of well bore 11 in oil bearing region 26, casing 23 includes perforations 27 that allow hydrocarbons and other material to enter annulus 28 between casing 23 and tubing 24. Gas is permitted to separate from the liquid products and travel up the annulus where it is captured. Liquid well products collect around pump barrel 29, which contains standing valve 30. Plunger 19 includes traveling valve 31. During the down stroke of the plunger, traveling valve is opened and product in the pump barrel is forced into the interior of tubing 24. When the pump begins its upstroke, traveling valve 31 is closed and the material in the tubing is formed forced up the tubing by the motion of plunger 19.

In deep wells the long sucker rod has considerable stretch, distributed mass, etc., and motion at the pump end may be radically different from that imparted at the upper end. One method of determining pump performance involves plotting a curve of rod load versus displacement. The shape of the curve or “card” reflects the conditions which prevail downhole in the well. U.S. Pat. No. 3,343,409 describes a method for determining the downhole performance of a rod pumped well by measuring surface data, (the surface card) and computing a load versus displacement curve (a “pump card” for the sucker rod string at any selected depth in the well).

Examples of pump cards, also referred to as downhole cards, are shown in FIG. 2. The plot of load versus position shows the operating conditions of the well during a full pump stroke. In broad terms, the corners of the top of the plot show the standing valve opening and closing points and the bottom shows the travelling valve opening and closing points. The shape of the plot, particularly the right edge can be indicative of how completely the pump is filling during a full stroke. The plot can be used to alter the operating conditions of the well, such as by slowing down the rod cycle to increase the time for the pump to fill during a rod cycle.

One problem with determining the plot of the pump card is that the work done at the surface is not transmitted directly to the pump as forces decrease the energy of the system. In order to have an accurate pump card compensation for losses in the system must be considered.

BRIEF SUMMARY OF THE INVENTION

One embodiment under the present disclosure comprises a method for calculating a downhole card for an oil and gas well. The method includes setting a damping factor to an initial value, and calculating a downhole card using the initial load value. The method also includes determining whether the difference between a true load value and an effective load value is within a desired error tolerance. Where the difference between the true load value and the effective load value is outside the desired error tolerance, the method can also include setting a new damping factor to a value equal at a mid-point between the initial damping factor and a damping factor bound. The method can also include recalculating the downhole card using the new damping factor, and determining whether the difference between a new true load value and a new effective load value is within a desired error tolerance. Where the difference between the new true load value and the effective load value is outside the desired error tolerance, the method can also include setting the new damping factor to a value equal at a mid-point between the previous damping factor and a damping factor bound.

Another embodiment comprises a method for calculating a downhole card for an oil and gas well. The method includes setting a first damping factor, calculating a first downhole card using the first damping factor, and determining whether a first difference between a first truck load value and a first effective load value is within a predetermined error tolerance. In response to determining that the first difference is outside the predetermined error tolerance, the method can also include setting a second damping factor at a midpoint between the first damping factor and a damping factor bound. The method can also include calculating a second downhole card using the second damping factor.

A further embodiment can comprise a method for generating a downhole card for an oil and gas well. The method includes setting an upstroke damping factor, setting a downstroke damping factor, and generating a downhole card using the upstroke damping factor and the downstroke damping factor. The method also includes calculating a top of stroke load, calculating an average upstroke load between a standing valve opening point and a standing valve closing point, calculating a downstroke load at a traveling valve opening point, and calculating an average downstroke load between the traveling valve opening point and a traveling valve closing point. The method further includes performing a primary convergence test, including calculating a horsepower difference between a pump horsepower and a hydraulic horsepower, and determining whether the horsepower difference is within a first predetermined error tolerance. In response to determining that the horsepower difference is outside the first predetermined error tolerance, the method can also include performing a secondary convergence test, including calculating an upstroke load difference between the top of stroke load and the average upstroke load, calculating a downstroke load difference between the downstroke load and the average downstroke load, determining whether the upstroke load difference is within a second predetermined error tolerance, and determining whether the downstroke load difference is within a third predetermined error tolerance. In response to determining that the upstroke load difference is outside the second predetermined error tolerance and/or that the downstroke load difference is outside the third predetermined error tolerance, the method can further include performing a bisection iteration, wherein the bisection iteration is repeated until the horsepower difference is within the first predetermined error tolerance, the upstroke load difference is within the second predetermined error tolerance, and the downstroke load difference is within the third predetermined error tolerance.

The foregoing has outlined rather broadly the features and technical advantages of the present invention in order that the detailed description of the invention that follows may be better understood. Additional features and advantages of the invention will be described hereinafter which form the subject of the claims of the invention. It should be appreciated by those skilled in the art that the conception and specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims. The novel features which are believed to be characteristic of the invention, both as to its organization and method of operation, together with further objects and advantages will be better understood from the following description when considered in connection with the accompanying figures. It is to be expressly understood, however, that each of the figures is provided for the purpose of illustration and description only and is not intended as a definition of the limits of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention, and, together with the general description of the invention given above, and the detailed description of the embodiments given below, serve to explain the principles of the present invention:

FIG. 1 depicts a diagram of a typical sucker rod pump used in oil wells;

FIG. 1A depicts a magnified view of area 1A in FIG. 1;

FIG. 1B depicts a magnified view of area 1B in FIG. 1;

FIGS. 2A-2E depict examples of pump cards;

FIG. 3 depicts a diagram of an example of a method to improve downhole card calculations;

FIGS. 4A-4F depict examples of pump cards using the method shown in FIG. 3; and

FIGS. 5A-5C depict comparisons of pump cards using the method shown in FIG. 3 to pump cards using a fixed damping factor.

The drawings are not intended to be limiting in any way, and it is contemplated that various embodiments of the invention may be carried out in a variety of other ways, including those not necessarily depicted in the drawings. The accompanying drawings incorporated in and forming a part of the specification illustrate several aspects of the present invention, and together with the description serve to explain the principles of the invention; it being understood, however, that this invention is not limited to the precise arrangements shown.

DESCRIPTION OF THE INVENTION

There are three forces coming into play when using rod pumping: elasticity, viscous friction, and mechanical friction. Elasticity takes the form of stress waves traveling up and down the rod string at the speed of sound as the rod string material stretches and compresses with the cyclic motion of the pumping unit. Produced Fluid being lifted to the surface creates a viscous force on the outer diameter of the rod string opposing its movement. Finally, either due to well deviation, paraffin or other external factors, the rod string and couplings cause mechanical friction with its contact with the inner diameter of the tubing.

In order to effectively control rod pumps, one must calculate not only the work done at the surface but also anticipate the energy losses to the system due to the above three conditions. The one-dimensional damped wave equation is used to calculate position and load at the pump using position and load measured at the surface.

The 1D damped wave equation reads:

$\begin{array}{cc}{v}^2\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}u}{\partial t^2}+D\frac{\partial u}{\partial t},& \left(1\right)\end{array}$

where the acoustic velocity is given by:

$v=\sqrt{\frac{144\mathrm{Eg}}{\rho }}.$

In the above equation energy is removed from the system using the damping term D∂u/∂t, to mimic the effects of energy losses due to viscous friction from produced fluids. The concepts described herein present a method to accurately determine and remove viscous friction from the downhole data.

Referring now to FIG. 3, a diagram of a method to improve downhole card calculations is shown. The method begins by initializing and setting the upstroke damping factor (Dup) to 0.25 and the downstroke damping factor (Ddn) to 0.5. The downhole card is then calculated using the initial values for Dup and Ddn. The pump horsepower (PMP_HP) can then be calculated using Riemann Sums where the pump horsepower is equal to the area of the downhole card. The standing valve opening (SVO), standing valve closing (SVC), traveling valve opening (TVO) and traveling valve closing (TVC) points are then calculated as are the values for F0UP_TRUE, F0DN_TRUE, F0UP_EFF, F0DN_EFF and F0_TRUE where F0UP_TRUE is the top of stroke load, F0DN_TRUE is the load at the TVO, F0UP_EFF is the average of load values between SVO and SVC, F0DN_EFF is the average of the load values between TVO and TVC and F0_TRUE is the fluid load. The difference between F0UP_TRUE and F0UP_EFF is calculated (DIFF_F0UP) as well as the difference between F0DN_TRUE and F0DN_EFF is calculated (DIFF_F0DN).

Next, the pump intake pressure (PIP) is calculated as is the production rate and hydraulic horsepower (HYD_HP). The primary convergence is tested by subtracting the PMP_HP from the HYD_HP. If that difference is less than an error tolerance ∈₁ then the downhole card calculation is sufficient. If the error tolerance is greater than ∈₁ then a secondary convergence is checked, namely |DIFF_F0UP|<∈₂ and |DIFF_F0DN|<∈₃. If the secondary convergence test is outside the error margins ∈₂ and ∈₃ the method moves to the bisection iteration method shown in FIG. 3.

The secondary convergence compares the TRUE fluid load lines with the EFFECTIVE fluid load lines. If the difference between True and Eff is less than a tolerance for both the upstroke and downstroke simultaneously, the downhole card calculation is sufficient. If not iterative bisection method is started. The bisection method involves determining the direction for each of the upstroke and downstroke damping factor using the sign of DIFF_F0DN and DIFF_F0UP, respectively. A new damping factor Dup is determined by the midpoint of the range between the old damping factor and the damping factor bound. A new damping factor Ddn is also determined by finding the midpoint of the old damping factor and the damping factor bound.

The new damping factors Dup and Ddn are then used to generate a new downhole card and the calculations described above are repeated and the error tolerances €1, ∈₂ and ∈₃ are checked again. If the tolerances are still outside the desired ranges, new Dup and Ddn damping factors are chosen using the midpoints between the previous Dup and Ddn values and the bounds. The process is repeated until all of the error tolerances ∈₁, ∈₂ and ∈₃ are within limits.

FIGS. 4A through 4F show the iterative process described above. FIG. 4A shows the initial pass with the default values for Dup (0.25) and Ddn (0.5). In the first iteration shown in FIG. 4B, the values are set to Dup=0.625 and Ddn=0.75 using the bisection method described as the error factors ∈₂ and ∈₃ from the initial pass are outside the error margins. In the second iteration, shown in FIG. 4C, Dup is unchanged from FIG. 4B, as error factor ∈₂ is within the error margin after the first iteration, but Ddn is set to 0.875 as error factor ∈₃ is outside the error margin. The iteration process continues in FIGS. 4D and 4E until the final downhole card is produced in FIG. 4F where the error factors are all within desired ranges.

FIGS. 5A, 5B and 5C show comparisons of downhole cards using the bisection method to downhole cards using a fixed damping factor.

Although the present invention and its advantages have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims. Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, composition of matter, means, methods and steps described in the specification. As one of ordinary skill in the art will readily appreciate from the disclosure of the present invention, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein may be utilized according to the present invention. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps.

Claims

1. A method for calculating a downhole card for an oil and gas well, the method comprising: (a) setting a damping factor to an initial value;(b) calculating a downhole card using the initial load value;(c) determining whether the difference between a true load value and an effective load value is within a desired error tolerance;(d) where the difference between the true load value and the effective load value is outside the desired error tolerance, setting a new damping factor to a value equal at a mid-point between the initial damping factor and a damping factor bound;(e) recalculating the downhole card using the new damping factor;(f) determining whether the difference between a new true load value and a new effective load value is within a desired error tolerance; and(g) where the difference between the new true load value and the effective load value is outside the desired error tolerance, setting the new damping factor to a value equal at a mid-point between the previous damping factor and a damping factor bound.

2. A method for calculating a downhole card for an oil and gas well, the method comprising: (a) setting a first damping factor;(b) calculating a first downhole card using the first damping factor;(c) determining whether a first difference between a first true load value and a first effective load value is within a predetermined error tolerance;(d) in response to determining that the first difference is outside the predetermined error tolerance, setting a second damping factor at a midpoint between the first damping factor and a damping factor bound; and(e) calculating a second downhole card using the second damping factor.

3. The method of claim 2, further comprising determining whether a second difference between a second true load value and a second effective load value is within the predetermined error tolerance.

4. The method of claim 3, further comprising, in response to determining that the second difference is outside the predetermined error tolerance, setting a third damping factor at a mid-point between the second damping factor and the damping factor bound.

5. The method of claim 4, further comprising calculating a third downhole card using the third damping factor.

6. The method of claim 2, wherein the first and second damping factors include first and second upstroke damping factors, respectively.

7. The method of claim 6, wherein setting the first damping factor includes setting the first damping factor to 0.25.

8. The method of claim 2, wherein the first and second damping factors include first and second downstroke damping factors, respectively.

9. The method of claim 8, wherein setting the first damping factor includes setting the first damping factor to 0.5.

10. The method of claim 2, further comprising performing a primary convergence test prior to determining whether the first difference is within the predetermined error tolerance.

11. A method for generating a downhole card for an oil and gas well, the method comprising: (a) setting an upstroke damping factor;(b) setting a downstroke damping factor;(c) generating a downhole card using the upstroke damping factor and the downstroke damping factor;(d) calculating a top of stroke load;(e) calculating an average upstroke load between a standing valve opening point and a standing valve closing point;(f) calculating a downstroke load at a traveling valve opening point;(g) calculating an average downstroke load between the traveling valve opening point and a traveling valve closing point;(h) performing a primary convergence test, including: (i) calculating a horsepower difference between a pump horsepower and a hydraulic horsepower, and(ii) determining whether the horsepower difference is within a first predetermined error tolerance;(i) in response to determining that the horsepower difference is outside the first predetermined error tolerance, performing a secondary convergence test, including: (i) calculating an upstroke load difference between the top of stroke load and the average upstroke load,(ii) calculating a downstroke load difference between the downstroke load and the average downstroke load,(iii) determining whether the upstroke load difference is within a second predetermined error tolerance, and(iv) determining whether the downstroke load difference is within a third predetermined error tolerance; and(j) in response to determining that the upstroke load difference is outside the second predetermined error tolerance and/or that the downstroke load difference is outside the third predetermined error tolerance, performing a bisection iteration, wherein the bisection iteration is repeated until (A) the horsepower difference is within the first predetermined error tolerance, (B) the upstroke load difference is within the second predetermined error tolerance, and (C) the downstroke load difference is within the third predetermined error tolerance.

12. The method of claim 11, wherein the bisection iteration includes: (i) determining an upstroke direction for the upstroke damping factor,(ii) determining a downstroke direction for the downstroke damping factor,(iii) setting a midpoint of a range between the upstroke damping factor and an upstroke damping factor bound as a new upstroke damping factor,(iv) setting a midpoint of a range between the downstroke damping factor and a downstroke damping factor bound as a new downstroke damping factor, and(v) generating a new downhole card using the new upstroke damping factor and the new downstroke damping factor.

13. The method of claim 11, further comprising calculating the pump horsepower using Riemann Sums where the pump horsepower is equal to an area of the downhole card.

14. The method of claim 11, further comprising calculating the standing valve opening point.

15. The method of claim 11, further comprising calculating the standing valve closing point.

16. The method of claim 11, further comprising calculating the traveling valve opening point.

17. The method of claim 11, further comprising calculating the traveling valve closing point.

18. The method of claim 11, further comprising calculating a pump intake pressure.

19. The method of claim 11, wherein setting the upstroke damping factor includes setting the upstroke damping factor to 0.25.

20. The method of claim 11, wherein setting the downstroke damping factor includes setting the downstroke damping factor to 0.5.