Tapered stators in positive displacement motors remediating effects of rotor tilt

Abstract

Tapered stator designs are engineered in a positive displacement motor (PDM) power section to relieve stator stress concentrations at the lower (downhole) end of the power section in the presence of rotor tilt. A contoured stress relief (i.e. a taper) is provided in the stator to compensate for rotor tilt, where the taper is preferably more aggressive at the lower end of the stator near the bit.

Claims

1. A stator for use in a positive displacement motor (PDM) power section, comprising: a stator, the stator having an inlet and an outlet, the stator further having a length L between stator inlet and stator outlet, wherein Zn represents a stator position along L; the stator further having an internal elastomer liner such that the elastomer liner provides the stator with a stator internal surface, the stator internal surface having lobes formed therein, wherein the lobes define helical pathways in the stator internal surface, wherein zeniths of the lobes at Zn define a stator internal minor diameter DMINn, and nadirs of the pathways at Zn define a stator internal major diameter DMAJn, wherein (DMINn+DMAJn)/2 further defines a stator average diameter DAVEn at Zn; and a taper formed on the stator internal surface, the taper commencing at stator position Z1 where Z1 is at least about 0.67 L measured from the stator inlet, the taper ending at stator position Z3 where Z3 is about 1.0 L measured from the stator inlet, wherein DAVE3≥DAVE1+(0.03×(DMAJ1−DMIN1)/2); wherein the elastomer liner extends from at least stator position Z1 to stator position Z3; wherein the elastomer liner has a tensile stress in a range between about 250 psi and about 1000 psi at 25% elongation; wherein the elastomer liner further has a tensile stress in a range between about 400 psi and about 1200 psi at 50% elongation; and wherein the elastomer liner further has a tensile stress in a range between about 500 psi and about 1600 psi at 100% elongation.

2. The stator of claim 1, in which a rotor is received inside the stator to form a power section having at least one stage, wherein the power section has a pressure drop capability represented by ΔP, wherein ΔP is at least about 180 psi/stage.

3. The stator of claim 1, in which a rotor is received inside the stator to form a power section having at least one stage, wherein the power section has a pressure drop capability represented by ΔP, wherein ΔP is at least about 200 psi/stage.

4. The stator of claim 1, in which the taper transitions between stator position Z1 and stator position Z2, wherein Z2 is at 0.77 L as measured from the stator inlet, wherein DAVE2≥DAVE1+(0.015×(DMAJ1−DMIN1)/2)).

5. A stator for use in a positive displacement motor (PDM) power section, comprising: a stator, the stator having an inlet and an outlet, the stator further having a length L between stator inlet and stator outlet, wherein Zn represents a stator position along L; the stator further having an internal elastomer liner such that the elastomer liner provides the stator with a stator internal surface, the stator internal surface having lobes formed therein, wherein the lobes define helical pathways in the stator internal surface, wherein zeniths of the lobes at Zn define a stator internal minor diameter DMINn, and nadirs of the pathways at Zn define a stator internal major diameter DMAJn, wherein (DMINn+DMAJn)/2 further defines a stator average diameter DAVEn at Zn; and a taper formed on the stator internal surface, the taper commencing at stator position Z1 at about 0.67 L measured from the stator inlet, the taper ending at stator position Z3 at 1.0 L measured from the stator inlet, wherein DMAJ3≥DMAJ1+(0.03×(DMAJ3−DMAJ1)/2); wherein the elastomer liner extends from at least stator position Z1 to stator position Z3; wherein the elastomer liner has a tensile stress in a range between about 250 psi and about 1000 psi at 25% elongation; wherein the elastomer liner further has a tensile stress in a range between about 400 psi and about 1200 psi at 50% elongation; and wherein the elastomer liner further has a tensile stress in a range between about 500 psi and about 1600 psi at 100% elongation.

6. The stator of claim 5, in which a rotor is received inside the stator to form a power section having at least one stage, wherein the power section has a pressure drop capability represented by ΔP, wherein ΔP is at least about 180 psi/stage.

7. The stator of claim 5, in which a rotor is received inside the stator to form a power section having at least one stage, wherein the power section has a pressure drop capability represented by ΔP, wherein ΔP is at least about 200 psi/stage.

8. The stator of claim 5, in which the taper transitions between stator position Z1 and stator position Z2, wherein Z2 is at about 0.77 L as measured from the stator inlet, wherein DMAJ2=DMAJ1+(0.015×(DMAJ2−DMAJ1)/2)).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) For a more complete understanding of embodiments described in detail below, and the advantages thereof, reference is now made to the following drawings, in which:

(2) FIG. 1 is a schematic illustration of rotor behaviors on a continuum 10 of eccentric displacement of the rotor from its normal rotation orbit;

(3) FIG. 2A depicts a series of exemplary cross-section slices 21 of a power section 20 on which FEA is performed, and FIG. 2B depicts the model derived from FIG. 2A;

(4) FIG. 3 is a plot 30 from FEA of normalized rotor eccentricity vs. position along PDM length;

(5) FIGS. 4A and 4B are schematic illustrations depicting contact pressure distributions from rotor tilt in a standard PDM power section 40 (FIG. 4A) and in a power section with remediating taper 50 (FIG. 4B);

(6) FIGS. 5A and 5B illustrate advantages of tapered stator embodiments disclosed herein on which only the minor stator diameter is tapered;

(7) FIGS. 6A and 6B are longitudinal representations of a PDM power section with a 2-stage tapered stator deployed to compensate for rotor tilt, in which FIG. 6B has its scale exaggerated to emphasize relevant aspects;

(8) FIGS. 7A and 7B are sections as shown on 6B in which stator has taper deployed on the minor diameter only;

(9) FIGS. 8A and 8B are sections as shown on 6B in which stator has taper deployed on both major and minor diameters;

(10) FIGS. 9A and 9B are sections as shown on 6B in which stator has taper deployed on major diameter only;

(11) FIGS. 10A and 10B are schematic illustrations depicting more specific embodiments of tapered stators more generally described with reference to FIGS. 6A and 6B;

(12) FIGS. 11A and 11B illustrate testing protocols undertaken to measure the effects of rotor tilt on power section performance, in which FIG. 11A illustrates test stand 100 and FIG. 11B illustrates linear position transducer assemblies 107, 108;

(13) FIGS. 12A and 12B illustrate aspects of a further FEA plot 130 depicting normalized rotor eccentricity vs. position along PDM length;

(14) FIG. 13 is a yet further FEA plot 150 depicting normalized rotor eccentricity vs. position along PDM length;

(15) FIG. 14 is an orbital plot showing tested rotor eccentricity in a conventional power section; and

(16) FIGS. 15A and 15B are plots 160, 170 comparing tested rotor eccentricity vs. differential fluid operating pressures at top (uphole) and bottom (downhole) ends in a power section, in which rotor behavior in a conventional stator is depicted on FIG. 15A, and rotor behavior in a power section with a stator adjusted for rotor tilt is depicted on FIG. 15B.

DETAILED DESCRIPTION

(17) The following description of embodiments provides non-limiting representative examples using Figures, diagrams, graphs, plots, schematics, flow charts, etc. with part numbers and other notation to describe features and teachings of different aspects of the disclosed technology in more detail. The embodiments described should be recognized as capable of implementation separately, or in combination, with other embodiments from the description of the embodiments. A person of ordinary skill in the art reviewing the description of embodiments will be capable of learning and understanding the different described aspects of the technology. The description of embodiments should facilitate understanding of the technology to such an extent that other implementations and embodiments, although not specifically covered but within the understanding of a person of skill in the art having read the description of embodiments, would be understood to be consistent with an application of the disclosed technology.

(18) Reference is now made to FIGS. 2A through 15B in describing currently preferred power section embodiments including tapered stators for remediating rotor tilt. For the purposes of the following disclosure, FIGS. 2A through 15B should be viewed together. Any part, item, or feature that is identified by part number on one of FIGS. 2A through 15B will have the same part number when illustrated on another of FIGS. 2A through 15B. It will be understood that the embodiments as illustrated and described with respect to FIGS. 2A through 15B are exemplary. The scope of the inventive material set forth in this disclosure is not limited to embodiments illustrated and described herein, or to other specific deployments thereof.

Finite Element Analysis

(19) FIGS. 2A through 4B describe the results of finite element analysis (FEA) examining the effect of rotor tilt on a hypothetical power section. FIG. 2A depicts a series of exemplary cross-section slices 21 of a power section 20 on which FEA is performed to determine the rotor's normalized eccentric orbital displacement along the PDM's length when subjected to rotor tilt forces expected in a high fluid pressure leaky PDM with linear pressure drop applied. The eccentric orbital displacement is thus configured to simulate expected rotor tilt in a high power PDM.

(20) FIG. 2B shows the model derived from FIG. 2A. FIG. 2B illustrates power section 20 including stator tube 22, stator elastomer 23, rotor 24, and nominal (design) orbit of rotation of rotor center 25.

(21) FIG. 3 a plot of the normalized position of the rotor's center under load versus its respective position along the power section from inlet to outlet. FIG. 3 is a predictive plot from FEA work on the model of FIGS. 2A and 2B. As used in this disclosure, the terms “normalized position of the rotor's centerline”, or the “normalized eccentricity” of the rotor, refer to correcting the rotor position in FEA for small deflections of the stator tube in the FEA model. The FEA model was not characterized for an infinitely stiff stator tube. Correction, or “normalizing”, of the rotor position (eccentricity) was required in order to remove the effect of small stator tube deflections on the rotor position inherent in applying FEA forces to an overall power section model. The x-axis on plot 30 on FIG. 3 shows the position along the length of the power section. The scale represents a theoretical power section length in inches. Zero is at the inlet. The y-axis shows the normalized eccentricity of the rotor's center. FIG. 3 illustrates that the tilting slope in about the last 80″ (34%) of the entire 235″ profile is much steeper than in about the first 155″. Further, about the last 10″-35″ (4%45%) of this exemplary power section design has a much steeper tilting slope than the rest of the length. FIG. 3 validates that rotor tilt is most prevalent in a zone near the outlet (bottom end near the bit) where local fluid pressure imbalances are forcing the interference fits between rotor and stator lobes on the non-eccentric side to engage heavily.

(22) FIGS. 12A, 12B and 13 are similar predictive FEA plots to FIG. 3, again depicting FEA work on the model of FIGS. 2A and 2B. As such, FIGS. 12A and 12B illustrate aspects of a further FEA plot 130 depicting normalized rotor eccentricity (y-axis) vs. position along PDM length from inlet to outlet in inches (x-axis). FIG. 13 illustrates aspects of a yet further FEA plot 150 depicting normalized rotor eccentricity vs. position along PDM length.

(23) FIGS. 12A and 12B should be viewed together. FEA plot 130 on FIGS. 12A and 12B represents a more idealized version of FIG. 3. The transmission was characterized to be stiffer in FIG. 3 for FEA purposes. FIGS. 12A and 12B (plot 130) simulate rotor behavior with a less stiff transmission that is more likely to reflect actual downhole conditions. Two hard (stiff) rubber types were simulated on FIGS. 12A and 12B, plotted with different simulated pressure drops to assess corresponding rotor deflection behavior. Lines 131-134 on FIGS. 12A and 12B correspond to the various rubber stiffness/pressure drop plots. The legend on FIGS. 12A and 12B may be “decoded” as follows: 2× or 3× is a rubber stiffness parameter; 1580 psi is a pressure drop parameter; and 0.75 ext-xyz” etc. correspond to non-linear pressure drop functions. To summarize, the legend on FIGS. 12A and 12B corresponds to Table 1 below:

(24) TABLE-US-00001 TABLE 1 Line Legend number Description 2x, 1580 psi 131 Stiff rubber, linear pressure drop 3x, 1580 psi 132 Very stiff rubber, linear pressure drop 2x, 1580 psi, 0.75 ext-xyz 133 Stiff rubber, non-linear pressure drop A 2x, 1580 psi, 0.75 ext-xz 134 Stiff rubber, non-linear pressure drop B

(25) Plot 130 on FIGS. 12A and 12B reveals several aspects of rotor behavior worthy of note. Brackets 139 and 138 on FIG. 12A highlight the last (bottom end) 12 inches and 50 inches of the power section respectively, which correspond to about the last 0.2 to 1.5 stage lengths at the bottom end. Brackets 137 and 136 on FIG. 12A indicate that undesirable bend behavior happens near the bottom end, with normalized eccentricity (y-axis) falling sharply in the last 0.2 to 1.5 stage lengths of the rotor. Rotor tilt would be evident in this region, binding the rotor against stator lobe tips and increasing friction at interference fits. Further, referring to reference line 135 on both FIGS. 12A and 12B, highly undesirable behavior happens when normalized rotor eccentricity falls below 1.0. Normalized rotor eccentricity of 1.0 is the nominal design orbit where rotor lobes contact stator lobe tips as designed, usually with an interference fit. Normalized rotor eccentricity below 1.0 suggests that the rotor is binding heavily against the stator lobe tips, causing high friction and shear stress in the stator lobes. Such highly undesirable behavior below a normalized rotor eccentricity of 1.0 is further illustrated by brackets 140 and 141 on FIG. 12B where approximately the last 6 inches to 8 inches of power section length is below the threshold and would be severely affected by rotor tilt.

(26) FIGS. 12A and 12B further demonstrate that rotor tilt behavior is substantially unaffected by variations in rubber stiffness and pressure drops. With small differences, lines 113-134 on FIGS. 12A and 12B all show overall generally similar rotor behavior as rubber stiffness and pressure drop varies.

(27) FIG. 13 illustrates aspects of a yet further FEA plot 150 depicting normalized rotor eccentricity (y-axis) vs. position along PDM length from inlet to outlet in inches (x-axis). FIG. 13 differs from previous FEA plots in that the model was characterized with a more aggressive pressure drop in order to simulate performance at or near the power section's operating limit (or at stall conditions). Similar to plot 130 on FIGS. 12A and 12B, plot 150 on FIG. 13 depicts rotor behavior (line 151) in a power section with a nominal stator pitch of 33.5 inches. In comparison to FIGS. 12A and 12B, plot 150 on FIG. 13 shows that undesirable rotor tilt behavior happens further from the outlet of the power section as a result of the more aggressive pressure drop. Brackets 155 and 154 on FIG. 13 highlight the last (bottom end) 30 inches and 64 inches of the power section respectively, and brackets 153 and 156 indicate the sharp fall in normalized rotor eccentricity in those regions. Further, and similar to plot 130 on FIGS. 12A and 12B, reference line 152 on FIG. 13 denotes that highly undesirable behavior happens when normalized rotor eccentricity falls below 1.0.

(28) FIGS. 4A and 4B are schematic illustrations depicting contact pressure distributions from rotor tilt in a standard PDM power section 40 (FIG. 4A) and in a power section with remediating taper 50 (FIG. 4B). FIG. 4A illustrates the rotor tilt effect shown in FIG. 3. FIG. 4B illustrates conceptually the proposed remediation of the rotor tilt effect shown on FIG. 4A using stators with strategically-located engineered tapers.

(29) Each of FIGS. 4A and 4B show schematically the following common features:

(30) Rotor 41, 51;

(31) Stator 42, 52;

(32) Nominal rotor centerline 43, 53;

(33) Nominal rotor orbit of rotation 44, 54;

(34) Nominal rotor eccentricity 45, 55; and

(35) Plane of last fully-sealed stage 46, 56.

(36) Referring first to FIG. 4A, fluid pressure force vectors F exert an increasing force on rotor 41 into stator 42 in standard power section 40. Reactionary contact pressure force vectors C increase correspondingly in stator 42, causing friction buildup in stator 42. FIG. 4A further depicts rotor tilt, particularly downhole of the plane of the last fully-sealed stage 46.

(37) In contrast to stator 42 on FIG. 4A, power section 50 on FIG. 4B provides stator 52 with an engineered taper 57 to remediate the rotor tilt seen on FIG. 4A. Fluid pressure force vectors F on FIG. 4B are reduced on rotor 51, which effect in turn reduces reactionary contact pressure force vectors C in stator 52. The effect of taper 57 on FIG. 4B is thus to stabilize rotor 51 and normalize contact pressure between the rotor 5 land stator 52.

Disclosed Embodiments within the Scope of this Disclosure

(38) It will be understood that the various embodiments set forth in this disclosure are exemplary only, and do not limit the full scope of this disclosure. As noted above, this disclosure addresses the rotor tilt problem by providing a tapered stator that preferably includes an aggressive taper near the outlet end of the PDM. Contrary to some of the teachings of the prior art, this disclosure seeks to remediate rotor tilt generally with a tapered stator whose tapered geometry is selected to intentionally separate the rotor from the stator to relieve contact pressure (and associated friction and tear stress) between rotor and stator. This disclosure particularly seeks to intentionally taper the stator aggressively in a region near the outlet where the rotor tilt is particularly problematic. In some embodiments, the taper near the outlet provides a clearance fit rather than an interference fit with the rotor. In preferred embodiments, the clearance fit is much larger than seen or expected in the prior art.

(39) It is acknowledged that this solution will likely sacrifice power output of the PDM by creating intentional leaks at the rotor/stator contact. However, the rotor remains stable in its rotation. Rubber stress concentrations are relieved. Power transfer and rotor stability is optimized in hard rubber stator embodiments, especially at high fluid pressure.

(40) As noted, this disclosure describes tapers designed to offer clearance fits where rotor tilt is expected. In particular, this disclosure favors aggressive tapers with high clearance fits at the outlet end of the PDM where rotor tilt forces are also expected to be especially high. These designs are not suggested by the prior art. The prior art is primarily concerned with thermal expansion. The prior art discusses gentle tapers that will loosen interference fit but will nonetheless keep leakage to a minimum in order to maintain power. Some prior art references teach keeping rotor/stator contact with looser fits to accommodate thermal expansion. In direct contrast, this disclosure describes solutions for rotor tilt in which the stator is intentionally separated from contact with the rotor in order to controllably stabilize local fluid pressure and normalize rotor/stator contact pressure.

(41) Preferred embodiments of tapered stators per this disclosure provide a 2-stage taper to remediate rotor tilt. The scope of this disclosure is not limited to 2-stage tapers, however. FIGS. 6A and 6B are longitudinal representations of a preferred PDM power section embodiment with such a 2-stage tapered stator. The scale in FIG. 6B has been exaggerated in order to illustrate relevant aspects better. FIG. 6A is more to scale. FIG. 6A is primarily for orientation of FIG. 6B with its exaggerated scale. FIG. 6B depicts an untapered Zone A near the inlet. A first taper T1 is shown in Zone B on FIG. 6B. First taper T1 is less aggressive and functions primarily to accommodate thermal expansion and some rotor tilt. A second taper T2 is shown in Zone C on FIG. 6B. Second taper T2 is more aggressive than first taper T1, and functions primarily in Zone C to stabilize local fluid pressure and normalize rotor/stator contact pressure.

(42) The rotor is shown in a neutral position on FIGS. 6A and 6B. It will be appreciated that the purpose of FIGS. 6A and 6B is primarily to illustrate schematically the 2-stage taper on the stator. The rotor is shown in a neutral position because its actual position will vary according to the specific 2-stage taper embodiment deployed within the more general scope of FIG. 6B.

(43) Tapers T1 and T2 on FIGS. 6A and 6B are illustrated as linear. It will nonetheless be appreciated that the scope of this disclosure is not limited to linear tapers. Other embodiments may provide arcuate, geometric or logarithmic profiles, for example.

(44) In some embodiments, about 50% of the PDM's initial length from the inlet is untapered. The first taper stage of the 2-stage taper begins at about the halfway point of the PDM's length from the inlet towards the outlet. “About halfway” is selected in these embodiments because the maximum power output of a multistage power section can best be obtained by utilizing a single inference fit for at least 50% of the inlet side. A transition between the untapered portion and the first taper stage is desirable.

(45) The first taper stage may transition into the second taper stage at a point anywhere up to about 90% of the PDM's length from inlet to outlet. The second (and more aggressive) taper stage preferably begins at a point along the PDM's length in a range from about 10% length to about 50% length from the outlet. A taper fit of about 102% to about 120% of paradigm design eccentricity is desirable at the outlet. Stated differently, and with reference to description of FIG. 10A below, taper embodiments may preferably include a taper defined by:
Stator minor diameter+[about (0.05×eccentricity of design) to about (0.2×eccentricity of design)]
“Eccentricity of design” refers to the radius of the expected (design) orbital pathway of the center of the rotor absent any rotor tilt and in an untapered stator. The first and second tapers may be engineered back from such taper fit at the outlet. A transition between the first taper stage and the second taper stage is desirable.

(46) In other embodiments, rotor tilt may be remediated according to this disclosure by a power section whose stator minor diameter at outlet is larger than the nominal inlet diameter and is tapered back to the nominal (inlet) minor diameter over a length spanning the outlet to about the midpoint of the power section. In some embodiments, the stator minor diameter at outlet may be larger than the nominal inlet diameter by at least about 5% of the eccentricity (0.5×stator lobe height). In some embodiments, the stator minor diameter at outlet is larger than the nominal inlet diameter and is tapered back to the nominal (inlet) minor diameter over a length spanning the outlet to about 25% of power section length back from outlet. In some embodiments, the stator minor diameter at outlet is larger than the nominal inlet diameter and is tapered back to the nominal (inlet) minor diameter over a length spanning the outlet to about 10% of power section length back from outlet. In some embodiments, the stator minor diameter at outlet is larger than the nominal inlet diameter and is tapered back to the nominal (inlet) with more than one taper where the most aggressive taper occurs in about the last 5% of PDM length measured from outlet, or alternatively in about the last 10% of PDM length measured from outlet, or alternatively in about the last 25% of PDM length measured from outlet, or alternatively in about the last 50% of PDM length measured from outlet.

(47) In other embodiments, stator tapers may be further compensated for expected thermal expansion in a conventional cylindrical fit. In such embodiments, tapers may be first designed to remediate rotor tilt, and then adjusted further for expected thermal expansion by removing additional material from stator lobes. In some such embodiments, at least an additional 0.015 inches of stator lobe material may preferably be removed in popular sized PDMs.

(48) A further exemplary embodiment of a 2-stage tapered stator within the scope of this disclosure may be derived with reference to FIG. 3. It will be recalled from prior description that FIG. 3 is an FEA-based plot of the normalized position of the FEA rotor's center versus its respective position along the power section from inlet to outlet. FIG. 3 illustrates that the tilting slope in about the last 80″ (34%) of the entire 235″ profile contour length is much steeper than in about the first 155″. Further, about the last 10″-35″ (4%-15%) of this exemplary power section design has a much steeper tilting slope than the rest of the length. An exemplary design to remediate the rotor tilt shown on FIG. 3 might provide two different stator tapers corresponding to the different tilts observed. Working back from the outlet, the stator might provide an aggressive taper on the final 30″-35″ of the stator's length. The stator may then provide a less aggressive taper in the region from about 30″-35″ back from the outlet to about 80″ back from the outlet. The taper slope in the 30″-80″ region might be about 0.25 to about 0.5 of the taper slope in the 0″-30″ region. When the tapered fit is optimized, the eccentricity of the tapered regions better match the eccentricity of the deflected rotor at maximum power and stall conditions.

(49) In some embodiments, the stator taper may be deployed based on an average of major and minor diameters. Conventional stator geometry and nomenclature acknowledges that a conventional stator has a length L between stator inlet and stator outlet, wherein Zn represents a stator position along L. The conventional stator further provides an internal surface with lobes formed in the internal surface, wherein the lobes define helical pathways in the stator internal surface. Zeniths of the lobes at stator position Zn define a stator internal minor diameter DMINn, and nadirs of the pathways at stator position Zn define a stator internal major diameter DMAJn, wherein (DMINn+DMAJn)/2 further defines a stator average diameter DAVEn at Zn. In embodiments deploying the taper based on an average of major and minor diameters, the taper may commence at stator position Z1 at about 0.67 L measured from the stator inlet, and the taper may end at stator position Z3 at 1.0 L measured from the stator inlet, in which DAVE3≥DAVE1+(0.03×(DMAJ1−DMIN1)/2). In other embodiments deploying the taper based on an average of major and minor diameters, the taper may provide a transition between stator position Z1 and stator position Z2, in which Z2 is at about 0.77 L as measured from the stator inlet, and in which DAVE2≥DAVE1+(0.015×(DMAJ1−DMIN1)/2)).

(50) Preferred embodiments within the scope of this disclosure deploy the taper on the minor diameter of the stator. The minor diameter taper is contrary to the teachings of the prior art. The prior art is concerned with thermal expansion and/or bending in power sections, where a minor diameter taper would likely not be suitable to maintain a desired but relaxed rotor/stator contact.

(51) FIGS. 7A and 7B are sections as shown on FIG. 6B in embodiments in which tapers T1 and T2 on FIG. 6B are deployed on the stator minor diameter only (see broken lines at stator minor diameters on FIGS. 7A and 7B denoting taper). FIGS. 8A and 8B are sections as shown on FIG. 6B in embodiments in which tapers T1 and T2 on FIG. 6B are deployed on both the stator major and minor diameters (see broken lines at stator major and minor diameters on FIGS. 8A and 8B denoting taper). FIGS. 9A and 9B are sections as shown on FIG. 6B in embodiments in which tapers T1 and T2 on FIG. 6B are deployed on the major diameter only (see broken lines at stator major diameters on FIGS. 9A and 9B denoting taper). Tapers as illustrated on FIGS. 7A through 9B are all embodiments within the scope of this disclosure, although minor diameter tapering per FIGS. 7A and 7B are currently preferred embodiments. FIGS. 7A through 9B have the following common features: Rotor R; stator S; center of rotor C.sub.R; progressing cavity PC; elevated fluid pressure P+; and maximum fluid pressure P.sub.MAX.

(52) FIGS. 10A and 10B are schematic illustrations depicting more specific embodiments of tapered stators more generally described above with reference to FIGS. 6A and 6B. FIG. 10A illustrates schematically a more specific stator embodiment 80 with a single bottom end taper 86, 87. Taper 86, 87 is analogous to taper T2 by itself on FIG. 6B. As is preferred herein, taper 86, 87 on stator embodiment 80 on FIG. 10A is on stator minor diameter 82 only. Stator embodiment 80 also includes stator centerline 81, exit diameter 83, stator tube 84 and stator elastomer 85. The geometry of taper 86, 87 on FIG. 10A includes a first relief depth 88, a first relief length 89 and a stator relief depth SPD.

(53) Exemplary embodiments according to FIG. 10A may be characterized from among the following:

(54) Preferred—Exit diameter 83≥Minor diameter 82+about (0.05×eccentricity of design)

(55) More preferred—Exit diameter 83≥Minor diameter 82+about (0.1×eccentricity of design)

(56) Preferred for aggressive drilling—Exit diameter 83≥Minor diameter 82+about (0.15×eccentricity of design)

(57) Preferred—First relief length 89≥about 0.1×Stator pitch length, but≤about 2.0×Stator pitch length

(58) More preferred—First relief length 89≥about 0.2×Stator pitch length, but≤about 1.5×Stator pitch length

(59) Most preferred—First relief length 89≥about 0.5×Stator pitch length, but≤about 1.0×Stator pitch length

(60) The term “eccentricity of design” as used above refers to the radius of the expected (design) orbital pathway of the center of the rotor absent any rotor tilt and in an untapered stator.

(61) FIG. 10B illustrates schematically a more specific stator embodiment 90 with a double bottom end taper 95A, 95B, 96A, 96B. Taper 95A, 95B, 96A, 96B is analogous to tapers T1 and T2 on FIG. 6B. As is preferred herein, taper 95A, 95B, 96A, 96B on stator embodiment 90 on FIG. 10B is on stator minor diameter 92A only. Stator embodiment 90 also includes stator centerline 91, second diameter 92B, exit diameter 92C, stator tube 93 and stator elastomer 94. The geometry of taper 95A, 95B, 96A, 96B on FIG. 10B includes a second relief depth 97, a second relief length 98A, a first relief depth 98B, a first relief length 99 and a stator relief depth SPD.

(62) Exemplary embodiments according to FIG. 10B may be characterized from among the following:

(63) Preferred—Exit diameter 92C≥Minor diameter 92A+about (0.05×eccentricity of design) AND Second diameter 92B≤Minor diameter 92A+about (0.025×eccentricity of design)

(64) More preferred—Exit diameter 92C≥Minor diameter 92A+about (0.1×eccentricity of design) AND Second diameter 92B≤Minor diameter 92A+about (0.05×eccentricity of design)

(65) Preferred—First relief length 99≥about 0.1×Stator pitch length, but≤about 2.0×Stator pitch length, AND Second relief length 98A≥about 1.0×First relief length 99, but≤about 2.0×First relief length 99

(66) More preferred—First relief length 99≥about 0.2×Stator pitch length, but≤about 1.5×Stator pitch length, AND Second relief length 98A≥about 1.0×First relief length 99, but≤about 2.0×First relief length 99

(67) Most preferred—First relief length 99≥about 0.5×Stator pitch length, but≤about 1.0×Stator pitch length, AND Second relief length 98A≥about 1.0×First relief length 99, but≤about 2.0× First relief length 99

(68) As noted above, the term “eccentricity of design” as used above refers to the radius of the expected (design) orbital pathway of the center of the rotor absent any rotor tilt and in an untapered stator.

(69) FIGS. 5A and 5B further illustrate advantages of tapered stator embodiments disclosed herein on which only the minor stator diameter is tapered. Power section 60 on FIG. 5A and power section 70 on FIG. 5B have the following common features:

(70) Rotor 61, 71;

(71) Stator tube 62, 72;

(72) Stator elastomer 63, 73; and

(73) Nominal rotational orbit of rotor center 64, 74.

(74) Referring first to FIG. 5A, arrow 65 on power section 60 denotes that the centripetal force of rotor rotation forces the rotor 61 outwards and into stator elastomer 63 at operating speed. Arrow 66 denotes that forces from fluid pressure are wanting to lift rotor 61 off stator elastomer 63 and push back against arrow 65 at low fluid pressure and high operating RPM of rotor 61. Arrow 67 denotes that it is not ideal to reduce major diameter of stator via taper since by doing so, further rotor tilt would be encouraged. There would be less elastomer material at the major diameter, allowing arrow 65 to further push the rotor 61 off its nominal rotational orbit 64 and into the stator elastomer 63.

(75) FIGURE SB illustrates power section 70 in a near stall condition. Arrow 75 denotes that the centripetal force urging rotor 71 outwards tends towards zero as a stall condition approaches. At this point, arrow 76 denotes that the forces from fluid pressure become most effective at or near stall conditions to lift rotor 71 off stator material 73 and to push rotor 71 off its nominal rotational orbit 74 and into opposing lobes in stator elastomer 73. Stress concentrations will result in the opposing stator lobes as a result of the rotor tilt. Note the opposing lobes are at a stator minor diameter. Arrow 77 denotes that tapering at the stator minor diameter would thus be beneficial to reduce stress concentrations in stator lobe due to the rotor tilt.

(76) In summary, therefore, FIG. 5A illustrates that tapering the major diameter may not be ideal because to do so might encourage the rotor in yet further outward direction from its normal orbit of rotation. This would likely encourage rotor tilt rather than discourage it. Limiting the outward movement of the rotor is also important for rotor head connection clearance. Further, the rotor is constrained by the major diameter of the stator under low pressure and maximum rpm. This is desirable so that stator lobe tips do not experience excess loading in compression during rotor orbiting. Tapering the major diameter may create a stator lobe that is disadvantageously too high. Normal torsional reaction forces at low loads can tear a lobe that is too high. Combining excess orbit and high rotor speed can also tear the lobe root due to excess tensile stresses generated from torsional reaction forces.

(77) FIG. 5B illustrates that tapering the minor diameter leaves untapered stator valleys at the major diameter to help stabilize the rotor and deter further rotor tilt. By comparison, minor diameter tapering removes rubber material from stator lobes, which reduces the potential for heavy contact with the rotor lobes in the presence of rotor tilt.

(78) Reducing stator lobe height via minor diameter tapering also addresses the potential for stator lobe tearing during stall (or near stall) events. It was noted above that in some embodiments, the required rubber elongation to survive a stalling event is at least approximately 35% to 50% strain. Thus, in order for the power section to obtain sufficient service life and reliability in the presence of rotor tilt, a stress relieving feature (taper) is needed near the exit of the power section to obtain a factor of safety that reduces the strain to a level less than about 35% during stall conditions. This may be obtained by reducing the lobe height of the stator elastomer via minor diameter tapering starting from the outlet and extending to about 10%-50% PDM length from the outlet.

(79) In some embodiments, the minor diameter taper near the outlet may enlarge the stator diameter at the outlet by at least 10% greater than the eccentricity (½ lobe height) of the stator profile. Such embodiments will reduce rubber strain at or near the outlet, especially in cases of heavy rotor tilt.

(80) Preferred embodiments may thus deploy the taper based on measurements of major diameter only, being indifferent to minor diameter (which may be constant). Referring back now to the conventional stator geometry and nomenclature set forth above, taper embodiments based on major diameter only may commence at stator position Z1 at about 0.67 L measured from the stator inlet and end at stator position Z3 at 1.0 L measured from the stator inlet, in which DMAJ3≥DMAJ1+(0.03×(DMAJ3−DMAJ1)/2). In other embodiments deploying the taper based on major diameter only, the taper may provide a transition between stator position Z1 and stator position Z2, in which Z2 is at about 0.77 L as measured from the stator inlet, and in which DMAJ2=DMAJ1+(0.015×(DMAJ2−DMAJ1)/2)).

(81) In a similar manner, stator material with higher modulus such as hard rubber, plastic or metal can have a factor of safety calculated for the exit area of the power section where high rotor tilt is experienced. In the case of these high modulus materials, it is more appropriate to consider failure as the point where galling pressures are exceeded. For hard materials, galling and rapid material overheating/removal are the mechanisms for failure. In this case, an oversized stator minor diameter can be calculated based on a minor stator diameter modification that allows the rotor to bend and minimize stress concentrations a region spanning about 10%-50% PDM length from the outlet.

(82) Note also that although preferred embodiments of the disclosed designs favor hard rubber throughout for power output, the scope of this disclosure is not limited in this regard.

(83) In some embodiments of power sections including stators with tapers configured to remediate rotor tilt consistent with this disclosure, the tapered stator may include an elastomer liner having: (1) a 25% tensile modulus in a range between about 250 psi and about 1000 psi; (2) a 50% tensile modulus in a range between about 400 psi and about 1200 psi; and (3) a 100% tensile modulus in a range between about 500 psi and about 1600 psi. The scope of this disclosure is not limited in these elastomer liner modulus regards, however.

(84) High modulus materials need not be limited to hard elastomers. Plastic, metal and hybrid stators are also within the scope of this disclosure. Aggressive tapers near the outlet of the PDM are also needed when using plastic or metal materials. In hybrid material arrangements, the highest modulus material of the stator profile is used at the exit end of the power section. Many of the high modulus materials have very low thermal expansions and so tapers addressing rotor tilt may not require further fit adjustment for thermal expansion.

(85) When utilizing other high modulus material such as plastic or metal as the interface with a metal rotor, the galling pressure is a critical parameter that advantageously should not be exceeded. When driving the power section at high pressure or under stall conditions, a tapered exit contour is advantageous to relieve the interface pressure between the deflected rotor and minor diameter stator lobes.

(86) In some embodiments of power sections including stators with tapers configured to remediate rotor tilt consistent with this disclosure, the power section preferably has a pressure drop capability represented by ΔP, wherein ΔP is preferably at least 180 psi/stage, and more preferably at least about 200 psi/stage. As used in this disclosure, pressure drop capability (ΔP) is a performance specification for the power section, and is functionally derived from a combination measurement of the stator lobe stiffness and the design rotor/stator fit (i.e. interference fit) for the power section. The stator lobe stiffness is functionally derived from a combination measurement of the stator elastomer's Modulus and the “reinforcement” behind the elastomer portion of the stator (e.g. the evenwall position or the overall rubber thickness to the outer tube). As used in this disclosure, pressure drop capability (ΔP) is defined as a fluid pressure drop per stage that will cause a 25% loss in rotor RPM at 1% squeeze. “Squeeze” is defined as the reduction in stator lobe height caused by the stator lobe interference fit with the rotor lobe under normal design conditions. AP capability also bears on the “power section rating”: Length of power section/stage length no. of stages; and power section rating=No. of stages×ΔP capability.

Testing Protocols

(87) FIGS. 11A and 11B illustrate testing protocols undertaken to measure and validate the effects and remediation of rotor tilt on power section performance described in this disclosure. Note that the testing protocols described herein with reference FIGS. 11A and 11B are exemplary only, and the scope of testing available to assess rotor tilt per this disclosure is not limited to testing conceived and executed described below with reference to FIGS. 11A and 11B.

(88) FIG. 11A illustrates test stand 100. Test stand 100 is from a conventional dynamometer (“dyno”) testing apparatus in which a full-sized power section may be driven with water or drilling fluid, preferably in a flow loop. As is known, drilling fluid is pumped through the power section to drive the rotor under controlled conditions. Measurements of the power section's performance and behavior may be taken. Test stand 100 on FIG. 11B was configured to measure dynamic rotor tilt by measuring the rotor axis location at the top and bottom ends of the rotor as power section 104. The power section was mated to a motor bearing assembly 101 and clamped to test stand 100 at three (3) places: a first near the top (uphole) end (clamp 102); a second near the bottom (downhole) end (clamp 103); and a third at the motor bearing assembly (clamp 105. A threaded output shaft of the motor was attached to the dynamometer shaft, which provided adjustable rotational resistance via a multi-plate disc brake 106.

(89) As further shown on FIG. 11A, two (2) linear position transducer assemblies 107, 108 were located at either end of the power section. Linear position transducer assemblies 107, 108 were each configured to measure eccentric rotor movement (i.e. rotor eccentricity) at their respective locations in order to determine rotor tilt.

(90) FIG. 11B illustrates linear position transducer assemblies 107, 108 in more detail. Linear position transducer assemblies 107, 108 each provided two (2) transducers 109, 110, with transducer 109 positioned to measure eccentric rotor motion in an x-axis, and transducer 110 positioned orthogonally to transducer 109 to measure eccentric rotor motion in a y-axis. As shown on FIG. 11B, transducers 109, 110 were configured to detect/measure positions of cams 111, 112 respectively. Cams 111, 112 were positioned to contact/press against the cylindrical ends of the rotor. Spring bias between cams 111, 112 and the cylindrical ends of the rotor enabled continuous contact and measurement through the rotor's entire orbital rotation.

(91) Raw rotor positional data from transducers 109, 110 at each of linear position transducer assemblies 107, 108 were converted to polar coordinates that provided eccentricity values at instantaneous points in time as each end of the rotor as it rotated within the stator. Data was recorded at a frequency of 2000 Hz in order to obtain rotor positional data with high granularity through a range of rotor operating speeds and other test parameters.

Tests and Test Results

(92) Two separate power sections A and B were tested separately to record rotor tilt. Power section A was a conventional power section, nominal 5″ diameter, with a 5/6 rotor/stator lobe configuration and 6.0 effective stages. Power section A further provided a stator whose elastomer was Abaco's HPW product, a hard rubber with fiber reinforcement, whose 25% tensile modulus may be in a range between about 250 psi and about 1000 psi. Power section B was identical to power section A, except that the bottom (downhole) end of the stator on power section B was adapted with a taper configured to remediate rotor tilt. The taper in power section B's stator was consistent with tapered stator embodiments described in this disclosure whose bottom-end tapers are specified herein for remediating rotor tilt.

(93) Three test runs were performed on each of power section A and B, at 150, 250 and 350 gallons per minute drilling fluid flow rate. At each flow rate on each test run, the torque applied by the motor to the dynamometer was increased in incremental steps to create a range of differential pressures and pressure drops across the power section. The dynamometer monitored and recorded fluid pressure, flow rate, motor torque and motor speed continuously for all test runs. Rotor eccentricity was monitored and recorded continuously by linear position transducer assemblies 107, 108 for all test runs per description above with reference to FIGS. 11A and 11B.

(94) FIG. 14 is an orbital plot 180 showing tested rotor eccentricity in a conventional power section (power section A) in which rotor axis position is traced at the bottom (downhole) end (dark-shaded solid lines 181) and compared to top (uphole) orbital rotor path (light-shaded solid line 182) and expected (nominal) orbital rotor path per design (broken line 183). The center of plot 180 represents the center of the stator. The rotational axis on orbital plot 180 shows the rotational position of the center of the rotor within the stator at the moment a data point was recorded, shown in degrees of orbital rotation. The radial axis on orbital plot 180 shows the radial distance of the center of the rotor from the center of the stator at the moment a data point was recorded, shown in inches. Nominal radius for the power section on plot 180 is 0.235 inches.

(95) Lines 181, 182, 183 on plot 180 on FIG. 14 map the pathways of the rotor center at the power section positions indicated. The nominal orbital rotor path per broken line 183 represents the designed nominal pathway of the rotor center for an ideal rotor orbit. The top orbital rotor path per light-shaded solid line 182 represents the observed pathway of the rotor center at the top of the power section per the testing described above with reference to FIGS. 11A and 11B. Line 182 represents a typical data scatter for rotor eccentricity at the upper end of a conventional power section. Line 182 depicts smooth concentric bands of measured data points tightly grouped together, collectively not straying far from the nominal pathway per line 183.

(96) In contrast, the bottom orbital rotor path per dark-shaded solid lines 181 on plot 180 on FIG. 14 represents the observed pathways of the rotor center at the bottom of the power section, again per the testing described above with reference to FIGS. 11A and 11B. Lines 181 represent a typical data scatter for rotor eccentricity for the lower end of a conventional power section. Lines 181 depict unstable, nonconcentric bands of data points not grouped together, departing substantially from nominal pathway per line 183. Interestingly, lines 181 on FIG. 14 show the dynamic behavior of the rotor at the bottom end of the power section is even more errant from nominal than was predicted via FEA on FIGS. 12A, 12B and 13 described above. FIGS. 12A, 12B and 13 predicts rotor pathway incursions at the lower end of the power section as low as 0.95 eccentricity (where 1.0 eccentricity is defined as nominal per line 183 on FIG. 14). FIG. 14 shows comparable rotor pathway incursions at the lower end of the power section as low as 0.60 eccentricity, which will inevitably increase stresses on stator lobes at and near the outlet. In summary, the testing results plotted on FIG. 14 validate the theoretical and FEA work set forth in this disclosure identifying rotor tilt as a significant PDM performance issue that may be remediated using aggressive lower end stator tapers.

(97) FIGS. 15A and 15B depict plots 160 and 170 respectively. Plots 160 and 170 compare rotor behavior observed and measured in power section A and power section B, respectively, according to the testing described above with reference to FIGS. 11A and 11B. To recap, power section A (FIG. 15A) is a conventional power section, and power section B (FIG. 15B) is identical to power section A, except that the bottom (downhole) end of the stator on power section B is adapted with a taper configured to remediate rotor tilt. Plots 160 and 170 on FIGS. 15A and 15B each depict rotor eccentricity vs. differential fluid operating pressures for power sections A and B, respectively, as observed and measured per the testing described above with reference to FIGS. 11A and 11B. Data points 161 about median 163 on FIG. 15A and data points 171 about median 173 on FIG. 15B are data points measured at a bottom (uphole) end of the respective power sections A and B. Data points 162 about median 164 on FIG. 15A and data points 172 about median 174 on FIG. 15B are data points measured at a top (uphole) end of the respective power sections A and B. Differential operating pressure on FIGS. 15A and 15B is depicted on the x-axis in units of psi. Rotor eccentricity on FIGS. 15A and 15B is depicted on the y-axis in units of inches. Similar to FIG. 14, rotor eccentricity in inches represents the radial distance of the center of the rotor from the center of the stator at the moment a data point was recorded.

(98) FIG. 15A shows top end eccentricity increasing slightly with increased fluid pressure, depicting a top end eccentricity range 166 of about 0.23 inches to about 0.245 inches at low fluid pressure and a top end eccentricity range 166 of about 0.24 inches to about 0.255 inches at high fluid pressure. Top end eccentricity range 166 for power section A on FIG. 15A thus changes little with fluid pressure.

(99) The same is true for top end eccentricity range 176 for power section B on FIG. 15B. Top end eccentricity again increases slightly on FIG. 15B with increased fluid pressure, with a top end eccentricity range 176 of about 0.225 inches to about 0.235 inches at low fluid pressure and a top end eccentricity range 176 of about 0.235 inches to about 0.245 inches at high fluid pressure.

(100) FIG. 15B shows bottom end eccentricity decreasing with increased fluid pressure, depicting a bottom end eccentricity range 165 of about 0.18 inches to about 0.24 inches at low fluid pressure and a bottom end eccentricity range 165 of about 0.145 inches to about 0.22 inches at high fluid pressure. Bottom end eccentricity range 165 for power section A on FIG. 15B thus increases with increased fluid pressure, from about 0.06 inches at lower fluid pressure to about 0.075 inches at higher fluid pressure.

(101) Different behavior is observed on FIG. 15B for bottom end eccentricity range 175 on power section B. Bottom end eccentricity decreases again with increased fluid pressure on power section B on FIG. 15B, although not as sharply as the decrease in bottom end eccentricity with increased fluid pressure seen for power section A on FIG. 15A. Bottom end eccentricity range 175 for power section B on FIG. 15B is about 0.2 inches to about 0.24 inches at low fluid pressure, and about 0.165 inches to about 0.215 inches at high fluid pressure. Bottom end eccentricity range 175 for power section B thus increases with increased fluid pressure, from about 0.04 inches at lower fluid pressure to about 0.05 inches at higher fluid pressure. Increased fluid pressure thus has a lesser effect on bottom end eccentricity range 175 for power section B on FIG. 15B than the effect increased fluid pressure has on bottom end eccentricity range 165 for power section A on FIG. 15A. Further, overall bottom end eccentricity deviation is demonstrably greater for power section A on FIG. 15A as compared to power section B on FIG. 15B. Bottom end eccentricity range 165 for power section A is about 50% higher than bottom eccentricity range 175 for power section B at lower fluid pressures (about 0.06 inches for power section A vs. about 0.04 inches for power section B). Bottom eccentricity range 165 for power section A is also about 50% higher than bottom eccentricity range 175 for power section B at higher fluid pressures (about 0.075 inches for power section A vs. about 0.05 inches for power section B).

(102) The data described and compared above with reference to FIGS. 15A and 15B validate that power section B on FIG. 15B demonstrates improved performance in remediating rotor tilt over power section A on FIG. 15A. The taper in power section B's stator at or near the bottom end is engineered to be consistent with tapered stator embodiments described in this disclosure. It can be concluded that such taper embodiments described herein are effective to stabilize orbital rotation of the rotor in power section B, particularly at the lower end and/or in the presence of high differential fluid pressures.

Variations and Additional Considerations

(103) Tapered fit varies by length from outlet by a nonlinear function that starts with aggressive slope and then shallows. Nonlinear function may be selected from a geometric function (e.g. square function), a logarithmic function or a spline function

(104) Tapered fit varies by length from outlet by a linear function or step function in multiple pieces.

(105) Aggressive tapering near outlet combined with a shallow taper fit for thermal expansion fit only. Examples:

(106) 1. Inlet, 50% shallow taper, 25% straight (untapered), 25% aggressive taper, outlet.

(107) 2. Inlet, 75% shallow taper, 25% aggressive taper, outlet.

(108) Note also manufacturing considerations—have to be able to remove and disassemble injection mold ends.

(109) Although the inventive material in this disclosure has been described in detail along with some of its technical advantages, it will be understood that various changes, substitutions and alternations may be made to the detailed embodiments without departing from the broader spirit and scope of such inventive material.