Note: Descriptions are shown in the official language in which they were submitted.
CA 02895793 2015-06-29
STRESS CALCULATIONS FOR SUCKER ROD PUMPING SYSTEMS
BACKGROUND
Field of the Disclosure
Aspects of the present disclosure generally relate to hydrocarbon production
using
artificial lift, and, more particularly, to a technique for stress
calculations at any depth for
sucker rod pumping systems.
Description of the Related Art
To obtain production fluids (e.g., hydrocarbons), a wellbore is drilled into
the earth
to intersect a productive formation. Upon reaching the productive formation,
pumps can
be used in wells to help bring production fluids from the productive formation
to a
wellhead located at the surface. This is often referred to as providing
artificial lift, as the
reservoir pressure may be insufficient for the production fluid to reach the
surface on its
own (i.e., natural lift).
The production of oil with a sucker-rod pump is common practice in the oil and
gas
industry. An oil well generally comprises a casing, a string of smaller steel
pipe inside the
casing and generally known as the tubing, a pump at the bottom of the well,
and a string
of steel rod elements, commonly referred to as sucker rods, within the tubing
and
extending down into the pump for operating the pump. Various devices as are
well
known in the art are provided at the top of the well for reciprocating the
sucker rod to
operate the pump.
SUMMARY
The systems, methods, and devices of the disclosure each have several aspects,
no single one of which is solely responsible for its desirable attributes.
Without limiting
the scope of this disclosure, some features will now be discussed briefly.
After
considering this discussion, and particularly after reading the section
entitled "Detailed
Description," one will understand how the features of this disclosure provide
advantages
that include improved production for artificially lifted wells.
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CA 02895793 2015-06-29
Aspects of the present disclosure generally relate to hydrocarbon production
using
artificial lift, and, more particularly, to a technique for stress
calculations at any depth for
sucker rod pumping systems.
One aspect of the present disclosure is a method for determining stress along
a
sucker rod string disposed in a wellbore. The method generally includes
receiving, at a
processor, measured rod displacement and rod load data for the sucker rod
string,
wherein the sucker rod string comprises a plurality of sections; and
calculating stress
values at a plurality of finite difference nodes for at least one of the
plurality of sections
based, at least in part, on the measured rod displacement and rod load data,
Another aspect of the present disclosure is a system. The system generally
includes a sucker rod string comprising a plurality of sections disposed in a
wellbore; at
least one sensor configured to measure rod displacement of the sucker rod
string; at
least one sensor configured to measure rod loading of the sucker rod string;
and a
processor configured to calculate stress values at a plurality of finite
difference nodes for
at least one of the plurality of sections based, at least in part, on the
measured rod
displacement and rod load data.
Yet another aspect of the present disclosure is a computer-readable medium.
The
computer readable medium generally includes computer executable code stored
thereon
for receiving measured rod displacement and rod load data for a sucker rod
string,
wherein the sucker rod string comprises a plurality of sections; and
calculating stress
values at a plurality of finite difference nodes for at least one of the
plurality of sections
based, at least in part, on the measured rod displacement and rod load data.
To the accomplishment of the foregoing and related ends, the one or more
aspects comprise the features hereinafter fully described and particularly
pointed out in
the claims. The following description and the annexed drawings set forth in
detail certain
illustrative features of the one or more aspects. These features are
indicative, however,
of but a few of the various ways in which the principles of various aspects
may be
employed, and this description is intended to include all such aspects and
their
equivalents.
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BRIEF DESCRIPTION OF THE DRAWINGS
So that the manner in which the above-recited features of the present
disclosure
can be understood in detail, a more particular description, briefly summarized
above, may
be had by reference to aspects, some of which are illustrated in the appended
drawings.
It is to be noted, however, that the appended drawings illustrate only typical
aspects of
this disclosure and are therefore not to be considered limiting of its scope,
for the
disclosure may admit to other equally effective aspects.
FIG. 1 illustrates a reciprocating rod lift system with a control unit for
controlling the
pump in an effort to extract fluid from a well, in accordance with certain
aspects of the
present disclosure.
FIG. 2 is a flow chart illustrating example operations for determining stress
along a
sucker rod string disposed in a wellbore, in accordance with certain aspects
of the
present disclosure.
FIG. 3 illustrates a surface card for a first example well, in accordance with
certain
aspects of the present disclosure.
FIG. 4 illustrates a downhole card for the first example well, in accordance
with
certain aspects of the present disclosure.
FIG. 5 is table showing a rod configuration for the first example well, in
accordance
with certain aspects of the present disclosure.
FIG. 6 illustrates a surface card for a second example well, in accordance
with
certain aspects of the present disclosure.
FIG. 7 illustrates a downhole card for the second example well, in accordance
with
certain aspects of the present disclosure.
FIG. 8 is a table showing a rod configuration for the second example well, in
accordance with certain aspects of the present disclosure.
FIG. 9 is a table showing results of stress analysis for the second example
well, in
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CA 02895793 2015-06-29
accordance with certain aspects of the present disclosure.
FIG. 10 is a graph showing the results of stress analysis for the first
example well,
in accordance with certain aspects of the present disclosure.
FIG. 11 is a graph showing the results of stress analysis for the second
example
well from the table of FIG. 9, in accordance with certain aspects of the
present disclosure.
FIG. 12 is a table showing example interpolated results for the minimum
stress,
maximum stress, and maximum allowable stress for a given depth for the first
example
well, in accordance with certain aspects of the present disclosure.
FIG. 13 is an example table showing example interpolated results for the
minimum
stress, maximum stress, and maximum allowable stress for a given depth for the
second
example well, in accordance with certain aspects of the present disclosure.
FIG. 14 is a diagram of an example tapered string having a plurality of
tapers, in
accordance with certain aspects of the present disclosure.
DETAILED DESCRIPTION
Various aspects of the disclosure are described more fully hereinafter with
reference to the accompanying drawings. This disclosure may, however, be
embodied in
many different forms and should not be construed as limited to any specific
structure or
function presented throughout this disclosure. Rather, these aspects are
provided so that
this disclosure will be thorough and complete, and will fully convey the scope
of the
disclosure to those skilled in the art. Based on the teachings herein one
skilled in the art
should appreciate that the scope of the disclosure is intended to cover any
aspect of the
disclosure disclosed herein, whether implemented independently of or combined
with any
other aspect of the disclosure. For example, an apparatus may be implemented
or a
method may be practiced using any number of the aspects set forth herein. In
addition,
the scope of the disclosure is intended to cover such an apparatus or method
which is
practiced using other structure, functionality, or structure and functionality
in addition to or
other than the various aspects of the disclosure set forth herein. It should
be understood
that any aspect of the disclosure disclosed herein may be embodied by one or
more
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CA 02895793 2015-06-29
elements of a claim.
The word "exemplary" is used herein to mean "serving as an example, instance,
or
illustration." Any aspect described herein as "exemplary" is not necessarily
to be
construed as preferred or advantageous over other aspects.
Although particular aspects are described herein, many variations and
permutations of these aspects fall within the scope of the disclosure.
Although some
benefits and advantages of the preferred aspects are mentioned, the scope of
the
disclosure is not intended to be limited to particular benefits, uses, or
objectives. The
detailed description and drawings are merely illustrative of the disclosure
rather than
limiting, the scope of the disclosure being defined by the appended claims and
equivalents thereof.
Aspects of the present disclosure provide techniques for stress calculations
for
sucker rod pumping systems. This may allow well operators to accurately
monitor the
pump fillage and control the pump accordingly.
EXAMPLE ARTIFICAL LIFT SYSTEM
The production of oil with a reciprocating rod lift system 100 (e.g., sucker-
rod
pump system or rod pumping lift system), such as that depicted in FIG. 1, is
common
practice in the oil and gas industry.
FIG. 1 illustrates a reciprocating rod lift system 100 with a control unit 110
(e.g.,
including a rod pump controller or variable speed drive controller) for
controlling the rod
pump in an effort to extract fluid from a well, according to certain aspects
of the present
disclosure. Although shown with a conventional pumping unit in FIG. 1, the
reciprocating
rod lift system 100 may employ any suitable pumping unit.
The reciprocating rod lift system 100 is driven by a motor or engine 120 that
turns
a crank arm 122. Attached to the crank arm 122 are a walking beam 124 and a
horsehead 101. A cable 126 hangs off the horsehead 101 and is attached to a
sucker
rod 102 (e.g., a string of steel rod elements or a continuous rod string). The
sucker rod
102 is attached to a downhole rod pump 104 located within the wellbore 128. In
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CA 02895793 2015-06-29
operation, the motor 120 turns the crank arm 122 which reciprocates the
walking beam
124 which reciprocates the sucker rod 102.
In the reciprocating rod lift system 100, the rod pump 104 consists of a pump
barrel 106 with a valve 114 (the "standing valve") located at the bottom that
allows fluid to
enter from the wellbore, but does not allow the fluid to leave. The pump
barrel 106 can
be attached to or part of the production tubing 130 within the wellbore 128.
Inside the
pump barrel 106 is a close-fitting hollow plunger 116 with another valve 112
(the
"traveling valve") located at the top. This allows fluid to move from below
the plunger 116
to the production tubing 130 above and does not allow fluid to return from the
tubing 130
to the pump barrel 106 below the plunger 116. The plunger 116 may be moved up
and
down cyclically by the horsehead 101 at the surface via the sucker rod 102,
where the
motion of the pump plunger 116 comprises an "upstroke" and a "downstroke,"
jointly
referred to as a "stroke." A polished rod 118, which is a portion of the rod
string passing
through a stuffing box 103 at the surface, may enable an efficient hydraulic
seal to be
made around the reciprocating rod string. A control unit 110, which may be
located at the
surface, may control the system 100.
Typically, the reciprocating rod lift system 100 is designed with the capacity
to
remove liquid from the wellbore 128 faster than the reservoir can supply
liquid into the
wellbore 128. As a result, the downhole pump does not completely fill with
fluid on every
stroke. The well is said to be "pumped-off' when the pump barrel 106 does not
completely fill with fluid on the upstroke of the plunger 116. The term "pump
fillage" is
used to describe the percentage of the pump stroke which actually contains
liquid.
Being a positive displacement pumping system, rod-pump systems (e.g.,
reciprocating rod lift system 100) can reduce the bottom hole pressure to a
"near zero"
value. The foremost goal of rod pumping optimization is to match well
displacement to
inflow, which may be difficult if inflow is unknown or highly uncertain.
Uncertainty related
to inflow may lead to an overly conservative approach, for example, where the
system is
designed or operated such that the pump displacement is lower than the inflow,
such as
by continuous pumping. In this case, the rod lift system runs without any
problem and is
sometimes referred to as "optimized" operation, although the well production
is usually
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CA 02895793 2015-06-29
suboptimal and losing revenue. In another example, uncertainty related to
inflow may
lead to an overly aggressive approach, for example, where the system is
designed or
operated such that the pump displacement is higher than the inflow, such as by
intermittent pumping. In this case, the downhole pump and rod lift system
suffers from
issues such as fluid pound, pump-off, gas interference, and correspondingly
higher failure
rates due to incomplete pump fillage.
Preventing stress related failures is part of optimizing a sucker rod pumped
well
(e.g., similar to pumping system illustrated in FIG. 1). Tracking the maximum
and
minimum rod stress along with appropriate pump off control may ensure longer
life for the
rods and the entire sucker-rod pumping system. The rod string 102 may be a
straight
string or a tapered string. A tapered string includes multiple sections (e.g.,
"tapers")
having varying diameters. Each section, or taper, may include a plurality of
rod elements.
Traditionally, stress is only computed at the top of each taper; however,
stress failures
occur in areas of high friction and stress, which may not be near the top of
the taper and,
therefore, may be undetected by traditional stress computations.
Stress may be defined as a material's internal resistance per unit area when
an
external load is applied to it. Stress analysis is the practice of evaluating
the stress
distribution within a given material. In a rod string (e.g., such as rod
string 102) subjected
to axial tension, stress analysis may involve computing the average normal
stress. In the
ideal case, the rod string can be considered prismatic, for example, similar
to a straight
bar whose cross-section is uniform throughout its length. Although, in
practice, a rod
string may not be prismatic.
The yield strength of a material may be defined as the amount of stress at
which
the material will begin to undergo plastic deformation. The tensile strength
of a material
may be defined as the maximum stress the material can withstand, due to
pulling or
stretching, before the material fails or breaks. Even though the load applied
to a
component is usually well below the yield strength of the material the
component is made
of, the component may eventually fail through many repeated loads. For
example, a
small amount of damage may be done to the component with each cycle which,
alone,
may be insufficient to cause the component to fail, but over repeated cycles
may
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CA 02895793 2015-06-29
=
accumulate and eventually cause a fatigue failure of the component.
The service factor is a factor used to account for the corrosiveness of the
environment. The value for the service factor is typically found between zero
and one,
although in some case the service factor may be greater than one.
Yield strength and tensile strength may be known quantities, and therefore,
for a
rod string operating under normal conditions, the lifetime for that rod string
may be known
or predetermined. Unfortunately, due to deviation, corrosion, paraffin build
up, and/or
casing collapse, the normal conditions may be ideal conditions rather than
actual
conditions. Instead, failures and rod life may be anticipated by closely
monitoring the
behavior of the stress function with respect to each other throughout the rod
string.
Some sections of the rod string may be more sensitive than others and,
therefore, a more
detailed and in-depth analysis may be used for those sections.
Sucker-rod pumps (e.g., similar to the reciprocating rod lift system 100
illustrated
in FIG. 1), may experience two types of failures: tensile failures and fatigue
failures. A
tensile failure may occur when the sucker rod is over-stressed, for example,
when the
force exerted on the sucker rod material results in an axial pulling force
overcoming the
tensile strength of the material. For example, if excessive pull is applied to
the sucker
rod, the rod stress may exceed the rod material tensile strength causing a
tensile break.
Tensile failures typically occur in the rod body, where the cross-sectional
area may be the
smallest. Tensile failures may materialize as a permanent stretch and/or small
breaks in
the sucker rod. Once the sucker rod has incurred tensile failures, if the
sucker rod is run
again, the failure points may become stress raisers since the load bearing
cross-sectional
area is reduced.
Although static loads may be tensile loads, fatigue failures may occur from
repeated load variations within the rod string. For example, ignoring inertial
effects and
rod buoyancy, on the downstroke, the rod load is equal to the rod weight,
while on the
upstroke, the rod load is equal to the rod weight and the fluid load. Thus the
fluid load
represents an alternating stress, which may lead to fatigue damage and/or
failure.
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Sucker rod failures are typically attributed to fatigue breaks, which may
occur at
stress levels well below the ultimate tensile strength or even below the yield
strength of
the sucker rod (e.g., made of steel material having a high tensile strength).
Repeated
stresses cause material fatigue or plastic tensile failure of the sucker rod.
The failure
may start at a stress raiser on the surface of the rod (e.g., which may be due
to tensile
failures caused by the repeated stresses). The incurred crack may progress in
a
direction perpendicular to the stress across the sucker rod, therefore
reducing the cross-
sectional area capable of carrying the load, at which point the rod breaks.
Rod failure may be prevented or reduced using stress analysis. Accordingly,
what
is needed are techniques and apparatus for stress analysis in sucker rod
pumping
systems.
An approach is provided herein for an in-depth, step by step, stress analysis
of
each section (e.g., each taper) for sucker rod pumping systems, such as the
system 100
illustrated in FIG. 1, using finite differences to solve the wave equation and
interpolation
of stress data in order to approximate stress at any level along the rod
string.
EXAMPLE STRESS CALCULATIONS FOR SUCKER ROD PUMPING SYSTEMS
Avoiding the creation of stress raisers caused by mechanical damage, corrosion
action, and/or rod wear is desirable to prevent rod failure. The definition of
the fatigue
endurance limit for any material, pertaining to steel rods, is the maximum
stress level at
which the steel can sustain cyclic loading conditions for a minimum of ten
million cycles.
Additionally, changes in cross-sectional area may create areas of higher local
stress.
Maintaining rod stress within safe limits may help prevent rod failure.
Techniques and apparatus are provided herein for stress analysis, enabling
stress
computation at any level down the rod string, for example, by using finite
differences and
polynomial interpolation.
FIG. 2 is a flow chart illustrating example operations 200 for determining
stress
along a sucker rod string disposed in a wellbore, in accordance with certain
aspects of
the present disclosure. The operations 200 may be performed by a processor
(e.g.,
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CA 02895793 2015-06-29
control unit 110). The operations 200 may include, at 202, receiving measured
(e.g.,
using one or more sensors) rod displacement and rod load data for the sucker
rod string
(e.g., rod string 102), wherein the sucker rod string comprises a plurality of
sections (e.g.,
tapers).
Optionally, at 204, a plurality of finite difference nodes may be selected
such that
the selected finite difference nodes have a uniform spacing in at least one of
the plurality
of sections. As will be discussed in more detail below, the finite difference
nodes may be
selected such that the uniform spacing satisfies a stability condition.
At 206, stress values may be calculated at the plurality of finite difference
nodes
for the at least one of the plurality of sections based, at least in part, on
the measured rod
displacement and rod load data. As will be discussed in more detail below, the
stress
values may be calculated using the Modified Everitt-Jennings algorithm for the
selected
finite difference nodes.
Optionally, at 208, the calculated stress values at the plurality of finite
difference
nodes may be interpolated (e.g., using cubic spline interpolation) to
determine stress
values (e.g., minimum stress, maximum stress, and/or maximum allowable stress)
at one
or more points on the at least one section of the sucker rod string (e.g., at
any depth).
Optionally, at 210, the interpolated stress values may be output to a display
(e.g.,
connected with the control unit 110). For example, for a given depth, the
interpolated
stress values for minimum stress, maximum stress, and maximum allowable stress
may
be displayed (e.g., as shown in FIGs. 12 and 13).
Optionally, at 212, one or more pump parameters of a rod pump system (e.g.,
stroke speed, stroke length, minimum rod load, or maximum rod load) comprising
the
sucker rod string (e.g., reciprocating rod lift system 100) may be adjusted
based, at least
in part, on the interpolated stress values.
Example Modified Everitt-Jennings Algorithm
In order to diagnose and control a rod-pumped well, the behavior of the rod
string
may be simulated (e.g., calculated). One method to control a well is based on
fillage
CA 02895793 2015-06-29
calculated from a downhole card. Downhole data can be directly measured by a
downhole dynamometer or can be calculated by solving the one-dimensional
damped
wave equation. However, calculating downhole conditions from measured surface
data
may be difficult because irreversible energy losses may occur along the rod
string due to
elasticity.
The irreversible energy losses may take the form of stress waves traveling
down
the rod string at the speed of sound. The one-dimensional damped wave equation
may
be used to model the propagation of stress waves in an ideal slender bar.
Thus,
considering the rod string to be prismatic, downhole conditions may be
correctly
calculated from the surface data using the one-dimensional damped wave
equation.
According to certain aspects, the modified Everitt-Jennings algorithm uses
finite
differences to solve the wave equation in order to model the behavior of the
rod string.
As part of the algorithm, an iteration on damping, a fluid load line
calculation, and a pump
fillage calculation are used in an effort to ensure that the downhole data is
as accurate as
possible.
The rod displacement of position x at time t, may be given by Equation 1:
u = u(x, t)
(Eq. 1)
Acoustic velocity (e.g., the velocity of sound in the rod string (ft/sec)) may
be given
by Equation 2:
\1144Eg
v = (Eq. 2)
where E is Young's modulus of elasticity (psi), g is the gravity constant
(32.2 (Ibm-ft)/(lbf-
sec2)), and p is the density of the sucker-rod string (lbrnift3).
Using the rod displacement u, the acoustic velocity v, and a damping factor c,
the
condensed one-dimensional wave equation may be given by Equation 3:
2 a2u a2u du
v a7c2 -W-t7 (Eq. 3)
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In Eq. 3, only the damping force of viscous nature is considered. The damping
force may be a complex sum of forces acting in the direction opposing the
movement of
the sucker-rod string, such as fluid forces and mechanical friction acting on
the sucker-
rod string, couplings, and tubing. Coulombs or mechanical friction effects may
not be
considered because of their dependence on unknown factors, such as deviation
and
corrosion. However, the fluid forces may be approximated by the viscous forces
arising
in the annular space. For example, let A represent the sucker-rod string's
cross-sectional
area (in.2) and k represent the friction coefficient. In order to account for
varying rod
diameters, Eq. 3 may be expanded as shown in Equation 4:
1 0 EA P-tu (x,t) = ¨pA 32u (x,t) _ c pA au (x,t),
(Eq. 4)
ax2 144g at2 ` 144g at`
According to certain aspects, the Modified Everitt-Jennings algorithm may be
used
to solve the linear hyperbolic differential equation of Eq. 4. The Modified
Everitt-Jennings
method uses a finite difference model. For example, the Modified Everitt-
Jennings
algorithm may combine a finite difference engine to solve the wave equation
along with a
Pump Fillage Calculation (PFC), capable of computing accurate pump fillage
regardless
of downhole conditions and a Fluid Load Line Calculation (FLLC), which uses
calculus
and statistics to not only compute fluid load for a stroke, but also to
approximate the
amount of mechanical friction present in that particular stroke. The Modified
Everitt-
Jennings may incorporate an iteration on damping, using either single or dual
damping
factors. First-order-correct forward differences may be used as analogs for
the first
derivative with respect to time and second-order-correct central differences
may be used
as analogs for the second derivative with respect to time. A slightly
rearranged second-
order-correct central difference may used as the analog for the second
derivative with
respect to position to account for different taper properties.
Accordingly to certain aspects, the boundary conditions for Eq. 4 may be
obtained
directly from the surface position-versus-time and load-versus-time data.
Because only
the periodic solutions may be desired, initial conditions may not be used in
Eq. 4. Let N
represent the number of recorded surface data points and M be the total number
of finite
difference nodes along the rod string down the wellbore, such that the Mth
finite
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difference node may be the last point above the pump. Let [iimi represent the
vector of
finite difference nodes along the rod string. Let fj}1\11 represent the vector
of sample points
taken at the surface. Let (gpR}Ni be the discrete function for the surface
polished rod
position-versus-time data and let ffpR}A: be the discrete function for the
surface polished
rod load-versus-time data.
One of the advantages of using finite differences to solve the wave equation
is, as
mentioned above, space discretization (e.g., the formation of a mesh). In the
case of
sucker-rod pumps, a mesh may be created in both time and space. The i
direction is
positive downwards as the mesh progresses down the rod string, while the 3
direction is
taken to be the time increments between the surface data readings. This may
allow each
taper to be split into numerous smaller sections, which may be as short as a
few feet.
According to certain aspects, the finite difference analogs in Eq. 4 may be
replaced to
produce the following equations.
Equation 5 shows the resulting equation for
initialization:
For] =1,=== ,N : uoj = g pR,J (Eq. 5)
Equation 6 shows the resulting equation from Hooke's law:
For] = 1,".,N:u1,1 = fpR,J.Ax
EA
(Eq. 6)
where ax is the space between two finite difference nodes of a particular
taper (ft).
Equation 7 shows the resulting equation:
For i = 2,===,M: = ¨ (1EA f[a(1 + cat)] = ui j+i ¨ [a(2 + cat)
T-x)
() 6,EA (AEA1 no + a )
u
(Eq. 7)
x x) x
(11944Ag)+4-(11)44A - 1
where a = At2
g and Ax = -2 (Ax+ + ax-) and At is the time spacing between
2
each surface sampling point (sec.)
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Equations 8 and 9 show the resulting equations at the pump:
upurnmi = (1 + cAt) um_1,j+1 ¨ cAt = um_Lj + um_ii
(Eq. 8)
EA ,
Fpuinpi = TA; (aum ¨ 4um_1,i um_2 j)
(Eq. 9)
where F represents the load acting on the rod element.
According to certain aspects, the stability condition associated with the
above finite
difference diagnostic model may be given as shown in Equation 10:
Ax <
(Eq. 10)
vAt ¨
As shown in Eq. 10, the stability condition is satisfied when the ratio of the
distance
between the finite difference nodes¨which is a uniform distance¨to the product
of the
acoustic velocity in the rod sting and the sampling time is equal to or less
than 1.
According to certain aspects, using finite differences as a tool to solve the
wave
equation enables the creation of a mesh (e.g., a space and time
discretization) at each
finite difference node for which position, load and, therefore, stress may be
computed.
Aspects of the present disclosure provide an enhanced stress analysis using
the
Everitt-Jennings algorithm to calculate stress at any depth along a sucker rod
string
including uniform selection of the finite difference nodes and polynomial
interpolation of
stress calculations.
Example Uniform Selection of Finite Difference Nodes
As mentioned above, a tapered string includes multiple sections (e.g.,
"tapers"),
each section having a different outer diameter, which generally decreases with
increasing
depth in a wellbore. Each section, or taper, may include one or more
individual rod
elements having that particular diameter.
FIG. 14 is a diagram of an example tapered sucker rod string 1400 having a
plurality of sections, in accordance with certain aspects of the present
disclosure.
Although in FIG. 14, the tapered sucker rod string 1400 is shown as having
four tapers
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1402, 1402, 1404, and 1406, this is merely exemplary, and the tapered sucker
rod string
1400 could have more or less than four tapers. Tapers in the rod string 1400
may be of
different lengths and materials. The weight of the sucker-rod string is
distributed along its
length, meaning any rod element carries at least the weight of the rod
elements below.
Ideally, the rod string may be designed to take into account deviation and
corrosion, so
that the rod string provides operation without failure for a reasonable amount
of time
(e.g., ten million cycles). Rod string design may involve determining rod
size, lengths of
the individual taper section, and the rod material used.
Rod strings are subject to cyclic loading, which creates a pulsating tension
on the
rod string. During the upstroke, the rod elements carry the load of the
fluids, the dynamic
loads, and the friction forces, while on the downstroke the rod elements carry
the weight
of the rod elements below, this time without the dynamic loads and friction.
As
mentioned above, changes in cross-sectional area in the rod string create
areas of
concentrated local stress.
While designing or analyzing sucker-rod strings, the maximum tensile and
compressive stress to which the rod string may be subjected may be determined.
Depending on the rod string design, the rod string may be more or less
susceptible to
failure at different locations of the taper.
Before solving the wave equation for the downhole data, the rod string may be
divided into M finite difference nodes. The selection of the spacing for the
nodes may be
done per taper, as the properties of the tapers may vary. According to certain
aspects, in
an effort to provide a complete and thorough stress analysis, the finite
difference
elements, or nodes, may be selected in such a way that the Ax or spacing in
between
each node is of similar magnitude for each taper. For example, an initial
number of finite
difference nodes (per taper) may be selected to satisfy the stability
condition. The
minimum Ax for all tapers may then be used to compute the number of finite
difference
elements for the rest of the tapers to ensure a uniform mesh. The use of a
quasi-uniform
mesh may allow for a more detailed and practical analysis of the stress
functions.
CA 02895793 2015-06-29
Example Per-Taper Cubic Spline Interpolation of Rod Stress
As mentioned above, using finite differences to solve the wave equation may
enable for the computation of position, load, and stress at any level down the
rod string.
According to certain aspects, techniques for interpolating (e.g., cubic spline
interpolation)
the stress data are provided, so that a stress value can be output at any
level down the
rod string.
The stress data may be a series of taper-specific values. The progression of
the
stress data per-taper may be quasi-linear. In other words, the maximum tensile
stress
may occur at the bottom surface, while the maximum compression stress may
occur at
the top surface. The stress values may vary linearly from the top surface to
the bottom
surface. However, stress raisers along the rod may cause the stress values to
vary non-
linearly.
Various algorithms may be used for interpolating a discrete function: Taylor
polynomials, Lagrange, Divided Differences, etc. Taylor polynomials, when used
to
interpolate a polynomial function, agree closely with the given function at a
specific point,
but the best accuracy is only available near that point. It may be desirable
for
interpolation to provide an accurate approximation over the entire interval.
Another approach for interpolating a discrete function may involve piecewise
polynomial approximation. Using a piecewise polynomial approximation, the
interval is
split into several sub-intervals on which a different interpolating polynomial
is generated.
High-degree polynomials can have an oscillatory nature, which implies that
even the
slightest fluctuation over a portion of the interval could produce large
fluctuations over the
entire interval. The simplest piecewise-polynomial interpolation is piecewise
linear
interpolation. However, as mentioned above, the behavior of the stress data in
the event
of a stress raiser or a possible failure ceases to be linear. This implies
that using
piecewise linear interpolation may not be sufficiently accurate and could
potentially hide
the increased peak that would denote an area of high normal stress
concentration.
According to certain aspects, cubic spline interpolation may be used. Cubic
spline
interpolation may be used to approximate stress at each taper using only four
constants
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CA 02895793 2015-06-29
for each stress data point. One advantage of using a cubic spline
interpolation is that it is
relatively simple. Also, because cubic splines are third-degree polynomials,
cubic splines
are, therefore, continuously differentiable on the taper interval, providing a
continuous
second derivative. Hence, calculus methods may be used on the smooth cubic
spline
interpolant in order to search for possible stress raisers and imperfections
in the stress
data, which could in turn imply a future failure.
According to certain aspects, using the stress versus depth data, a cubic
spline
interpolant may be generated for each taper. The tridiagonal system generated
during
the cubic spline interpolation may then be solved, for example, using Grout
Factorization
or similar methods.
Downhole data may be computed for a particular stroke. The downhole data may
include N position values for each of the M finite difference nodes down the
rod string.
According to certain aspects, using the position values and Hooke's Law, N
load values
may be obtained corresponding to the N position values. Stress values may then
be
computed using the formula for uniform normal stress shown in Equation 11:
=-
(Eq. 11)
A'
where a represents the normal stress. Therefore, at any finite difference node
down the
rod string, the minimum stress, the maximum stress, and the maximum allowable
stress
may be computed. The minimum stress may be computed using the above equation
by
taking F = Emiõ, while the maximum stress may be computed by taking F = Fmax.
The
computation of maximum allowable stress may use the minimum stress and the
taper-
specific values of tensile strength and service factor.
The techniques described above may be used in an effort to ensure that the
downhole data obtained through the Modified Everitt-Jennings algorithm is as
accurate
as possible to actual downhole data. FIGs. 3 and 4 illustrate an example
surface card
300 and an example downhole card 400, respectively, for an example first well,
in
accordance with certain aspects of the present disclosure. FIG. 5 is a table
500 showing
the rod configuration for the first example well, in accordance with certain
aspects of the
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CA 02895793 2015-06-29
present disclosure. The first example well represents a deep well where steel
rod
elements (e.g., having a first tensile strength and service factor) combined
with sinker
bars are used. For each example, the stroke represents a full or near full
pump fillage
card
FIGs. 6 and 7 illustrate an example surface card 600 and an example downhole
card 700, respectively, for a second example well, in accordance with certain
aspects of
the present disclosure. FIG. 8 is a table 800 showing the rod configuration
for the second
example well, in accordance with certain aspects of the present disclosure.
The second
example well represents a shallower well than the first example well, and
grade KD rod
elements are used in the second example well, which would suggest a heavy-load
application in an effectively inhibited corrosive environment. In the second
example well,
the KD rod elements are AISI 4720 nickel-chromium-molybdenum alloy steel
(e.g.,
having a second tensile strength and service factor).
As shown in FIGs. 3-5 and FIGs. 6-8, the surface and downhole cards are
different
for the different wells, for example, due to different well conditions and
different types of
rods. FIG. 9 is an example table 900 showing results of stress analysis for
the second
well, in accordance with certain aspects of the present disclosure. The finite
difference
element distribution (third column in table 900) is displayed as related to
the depth (first
column in table 900) and taper number (second column in table 900). The
matching
values for the minimum stress, maximum stress and maximum allowable stress are
also
displayed in the fourth, fifth, and sixth columns, respectively, in table 900.
For the second
example well, the total number of finite difference elements M is 50. Although
not shown
in the figures, for the first example well, the total number of finite
difference elements M is
80.
FIG. 10 is a graph 1000 showing the results of stress analysis for the first
example
well, in accordance with certain aspects of the present disclosure. FIG. 11 is
a graph
1100 showing the results of stress analysis for the second example well from
the table
900, in accordance with certain aspects of the present disclosure.
According to certain aspects, the first taper 1002 (or 1102), second taper
1004 (or
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CA 02895793 2015-06-29
1104), third taper 1006 (or 1106), and fourth taper 1008 (or 1108) illustrated
in FIG. 10
(or FIG. 11), may correspond to the tapers 1402, 1404, 1406, and 1408
illustrated in FIG.
14. In FIGs. 10 and 11, the results of the stress analysis for the minimum
stress, the
maximum stress, and the maximum allowable stress are plotted against the
depth. The
marker points on each of the three curves represent the finite difference
nodes or
elements. As shown in FIG. 10, for the first example well, the minimum stress
starts at
10872 psi/in2 and decreases to 5204 psi/in2 for the first taper 1002. For the
second taper
1004, the minimum stress starts at 6674 psi/in2 decreasing to 1590 psi/in2.
For the third
taper 1006, the minimum stress decreases from 2383 psi/in2 to -2565 psi/in2.
For the
fourth taper 1008, the minimum stress decreases from 2390 psi/in2 to -2986
psi/in2.
As shown in FIG. 10, the maximum stress for the first example well varies in a
similar manner as the minimum stress. The maximum stress starts at 33964
psi/in2 and
decreases to 25599 psi/in2 for the first taper 1002. For the second taper
1004, the
maximum stress decreases from 33351 psi/in2 to 23572 psi/in2. For the third
taper 1006,
the maximum stress decreases from 32159 psi/in2 to 22839 psi/in2. For the
fourth taper
1008, the maximum stress decreases from 3691 psi/in2 to 2802 psi/in2.
As shown in the FIG. 10, the maximum allowable stress for the first example
well
varies in a similar manner as the minimum and maximum stress, but with
increased
amplitude. For the first taper 1002, the maximum allowable stress starts at
54061 psi/in2
and decreases to 51951 psi/in2. For the second taper 1004, the maximum
allowable
stress decreases from 52502 psi/in2 to 50596 psi/in2. For the third taper
1006, the
maximum allowable stress decreases from 50893 psi/in2 to 49037 psi/in2. For
the fourth
taper 1008, the maximum allowable stress decreases from 21155 psi/in2 to 20820
psi/in2.
As shown in the graph 1100 of FIG. 11, for the second example well, the
minimum
stress starts at 9596 psi/in2 and decreases to 5364 psi/in2 for the first
taper 1102. For the
second taper 1104, the minimum stress decreases from 6699 psi/in2 to 1518
psi/in2. For
the third taper 1106, the minimum stress decreases from 2059 psi/in2 to -1753
psi/in2.
For the fourth taper 1108, the minimum stress decreases from -1352 psi/in2 to -
2377
psi/in2.
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CA 02895793 2015-06-29
As shown in FIG. 11, the maximum stress for the second example well varies in
a
similar manner as the minimum stress. The maximum stress starts at 24511
psi/in2 and
decreases to 18654 psi/in2 for the first taper 1102. For the second taper
1104, the
maximum stress decreases from 24062 psi/in2 to 15596 psi/in2. For the third
taper 1106,
the maximum stress decreases from 21032 psi/in2 to 14379 psi/in2. For the
fourth taper
1108, the maximum stress decreases from 1872 psi/in2 to 693 psi/in2.
As shown in the FIG. 11, the maximum allowable stress for the second example
well varies in a similar manner as the minimum and maximum stress, but with
increased
amplitude. The maximum allowable stress starts at 25315 psi/in2 and decreases
to
23053 psi/in2 for the first taper 1102. For the second taper 1104, the maximum
allowable
stress decreases from 30892 psi/in2 to 28124 psi/in2. For the third taper 1106
the
maximum allowable stress decreases from 28413 psi/in2 to 26589 psi/in2. For
the fourth
taper 1108, the maximum allowable stress decreases from 26375 psi/in2 to 26042
psi/in2.
As shown in FIGs. 10 and 11, a jump occurs in the stress curves while
transitioning from one taper to the next; this jump is attributed to the
different taper
properties. The finite difference nodes along the rod string are quasi-uniform
to provide a
more thorough analysis of the rod string.
When analyzing the stress values for a rod string, it may be desirable to
maintain
the peak stress under the maximum allowable stress, while keeping the minimum
stress
as high as possible. The closer the maximum stress is to the maximum allowable
stress,
the more loaded the rod string is and, therefore, the shorter its failure-free
operation time
may be. As shown in FIG. 11, for the second well, the calculated peak stress
may be
close to the maximum allowable stress. This implies that the rod elements in
the first
taper 1102 are highly loaded, which in turn implies that these rod elements
may fail
sooner than what is expected when considering the yield strength and tensile
strength of
these rod elements.
Conventionally, values for minimum stress, maximum stress, and maximum
allowable stress are computed at the top of each taper. As shown in FIGs. 10
and 11,
under normal conditions, the stress values of greatest magnitude may occur at
the top of
CA 02895793 2015-06-29
the taper and decrease almost linearly until the end of the taper. However,
this
assumption might no longer be valid in a deviated well. Displaying stress only
at the top
of the taper might not encapsulate any stress variation arising from points of
high dogleg
severity. Also, considering that tapers can be thousands of feet long,
computing stress
values at only one point seems a simplistic approach. Thus, minimum stress,
maximum
stress, and maximum allowable stress may be interpolated using cubic splines
or another
polynomial interpolation in an effort to produce a smooth function to enable
the
computation of minimum stress, maximum stress, and maximum allowable stress at
any
depth down the rod string. According to certain aspects, in the case where the
desired
depth lies at the edge a taper, for example, after the last finite difference
node or before
the first finite difference node on any taper, piecewise linear interpolation
may be used to
interpolate stress values at that point.
FIGs. 12 and 13 are tables 1200, 1300 showing example interpolated results for
the minimum stress, maximum stress, and maximum allowable stress at a given
depth for
the first example well and the second example well, respectively. For both the
first
example well and the second example well, the stress results are interpolated
for a depth
of 1637 ft. For the first example well, this depth occurs in the first taper
1002, in between
finite difference element 18 and 19. As shown in table 1200, for the first
example well,
the interpolated values for minimum stress, maximum stress, and maximum
allowable
stress are 6016 psi/in2, 26949 psi/in2, and 52256 psi/in2, respectively. For
the second
example well, this depth occurs in the second taper 1004, in between elements
1 and 2.
As shown in table 1300, the interpolated values for minimum stress, maximum
stress and
maximum allowable stress are 6685 psi/in2, 24039 psi/in2, and 30885 psi/in2,
respectively.
The ability to compute stress values at any depth down the rod string may
allow
for improved management of downhole conditions such as deviation or corrosion.
When
the dogleg severity of the wellbore path is above a certain risk angle, it is
then possible to
focus on the stress values at that point and in the vicinity of that point,
providing an
improved picture of the loads and stresses for that particular rod section.
This may be
useful for anticipating rod failures.
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CA 02895793 2015-06-29
The techniques described above may rely on the accurate computation of
downhole data. When computing downhole data from surface data using the wave
equation, it is desirable to handle viscous damping properly. Additionally,
although the
wave equation assumes a vertical-hole model, often it is applied to wells
having non-
negligible deviation. The Modified Everitt-Jennings algorithm combines robust
iteration
on damping along with fluid load line computation capable of estimating the
presence of
mechanical friction in the well. This may ensure accurate downhole data
regardless of
downhole conditions.
The above stress analysis methodology, as part of the Modified Everitt-
Jennings
algorithm, may allow the user to monitor any section of the rod string. Stress
values can
be computed at every finite difference element, which can be spaced apart as
small as a
few feet, for example. Additionally, the capability to compute interpolated
stress values at
any depth may allow the user in-depth inspection of the stress distribution at
a certain
point or series of points, completing the stress analysis picture.
Using the above methodologies, it may be possible to more accurately
anticipate
the life of a rod string. Based on the in-depth stress analysis, and the
detailed results on
the loading of any particular section of the rod string, the user can
anticipate what the life
of a rod, taper, or installation is going to yield.
As used herein, the term "determining" encompasses a wide variety of actions.
For example, "determining" may include calculating, computing, processing,
deriving, and
the like. As used herein, a phrase referring to "at least one of" a list of
items refers to any
combination of those items, including single members.
The methods disclosed herein comprise one or more steps or actions for
achieving
the described method. The method steps and/or actions may be interchanged with
one
another without departing from the scope of the claims. In other words, unless
a specific
order of steps or actions is specified, the order and/or use of specific steps
and/or actions
may be modified without departing from the scope of the claims.
Any of the operations described above may be included as instructions in a
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CA 02895793 2015-06-29
computer-readable medium for execution by the control unit 110 or any other
processing
system. The computer-readable medium may comprise any suitable memory for
storing
instructions, such as read-only memory (ROM), random access memory (RAM),
flash
memory, an electrically erasable programmable ROM (EEPROM), a compact disc ROM
(CD-ROM), or a floppy disk.
While the foregoing is directed to aspects of the present disclosure, other
and
further aspects of the disclosure may be devised without departing from the
basic scope
thereof, and the scope thereof is determined by the claims that follow.
23
