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Analysis and Quantification of
Wellbore Tortuosity
Jon Bang, Onyemelem Jegbefume, Adrián Ledroz, and John Weston, Gyrodata Incorporated; and
Jay Thompson, SandRidge Energy
Summary
Small-scale wellbore tortuosity—variations in attitude on a length
scale smaller than standard survey intervals of 30 m (100 ft)—is
generally neglected because of its small effect on the final position of the well and its unclear relation to traditional dogleg severity (DLS). However, it is well-known that such tortuosity may
have significant influence on the drilling process and on drilling
efficiency. Furthermore, it is a crucial factor for the design and installation of completions and production equipment, because a
highly tortuous wellbore section, depending on borehole diameter
and tortuosity amplitude and along-hole distribution, may exert
strong bending forces on such equipment, or high friction on moving parts.
This paper describes a novel methodology for analyzing the
tortuosity of openhole wellbores, casing, drillstring, and production tubing. Several related tortuosity parameters are described,
and examples of application to field data are included. The methods use high-resolution survey data [measured depth (MD), inclination, and azimuth], which may, in principle, originate from any
surveying tool or service capable of providing such data. The
methodology requires input data from a single survey only. On
the basis of a user-defined length as single external parameter, tortuosity can be analyzed on any length scale greater than approximately 10 times the input survey-data interval, and with a
maximum resolution equaling twice the data interval. The processed parameters include relative elongation of a tortuous section
compared with a straight-line section, transverse displacements
from the straight line, and maximum available diameter for a
downhole device caused by the small-scale bendings of the section. The results can be displayed as graphs vs. MD, or as 3Drendered views of the actual wellbore or tubing shape.
Results from various field cases are included, in which the tortuosity analysis was applied to high-resolution continuous gyro
survey data collected in cased wellbores. In all the field cases, the
novel methods revealed sections of considerable tortuosity that
were either unnoticed, or located with unacceptably low accuracy,
by conventional methods. These results led to re-evaluation of the
planned locations for completion and production equipment.
The characterization of the wellbore in terms of tortuosity on
various length scales may be of crucial importance for the functionality and lifetime of permanently installed equipment. For
example, identification of highly tortuous sections will aid the
placement of rod-guide wear sleeves, increasing the rod and casing life and reducing the workover frequency. Another application
is the identification of low-tortuosity sections in which downhole
pumps or other equipment will not be subject to excess bending.
In addition, the tortuosity results may help evaluate the drilling
equipment and the drilling process.
Introduction
Definition of Tortuosity. Gaynor et al. (2001) defines wellbore
tortuosity as “the amount by which the wellbore deviates from the
planned trajectory.” Although this definition may be useful in
many applications, we find it somewhat narrow, because it limits
the evaluation of tortuosity to a comparison between the outcomes
of two well-construction phases: planning and drilling (including
surveying). It should be emphasized that tortuosity is purely a geometric property of the wellbore. Thus, the process that created the
tortuosity (drilling) is subordinate to this, and the reference should
be more basic than a planned trajectory. This leads us to propose
the following definition, which we will use throughout this paper:
“Tortuosity is any deviation from a straight line.”
This definition can be applied in two or three dimensions; furthermore, it implies that tortuosity may exist on any length scale,
as illustrated in Fig. 1.
Why Analyze Tortuosity? Small-scale tortuosity [i.e., tortuosity
occurring on intervals significantly shorter than a drillstring stand
(30 m)] has been difficult to quantify because, in general, high-resolution data are not collected in traditional surveys. Nevertheless,
such tortuosity is recognized as an important factor in the drilling
process (Chen et al. 2002; Samuel et al. 2005; Mason and Chen
2006). High levels of tortuosity may imply significantly increased
friction factors and increased torque-and-drag values (Skillingstad
2000) and excessive casing wear (Samuel and Gao 2013). In
severe cases, this may lead to stuck pipe or limit the distance of
extended-reach-drilling operations (Stuart et al. 2003). To analyze
and optimize the drilling process, advanced computer models
have been developed, which take into account the interactions
between bottomhole assembly (BHA), drillstring dynamics, fluid
and formation properties, and wellbore geometry (Liu et al. 2004;
Menand et al. 2009; Marck et al. 2014). Such models may predict
the resulting small-scale tortuosity for a particular drilling scenario (Marck and Detournay 2015), or they use tortuosity as one
of many input parameters (see the subsection on DLS).
Tortuosity may also affect the completion and production
phases of a well. The large-scale wellbore curvature is normally
designed for problem-free transportation and installation of downhole devices (Reid et al. 2013). However, unforeseen and possibly
undetected small-scale tortuosity may cause significant reductions
in the effective inner diameter of the wellbore (cased or openhole).
This may lead to stuck equipment, or bending moments exceeding
the manufacturer’s specifications, with reduced functionality or
lifetime as potential consequences. A concrete example is the use
of artificial-lift methods for the extraction of oil with a rod-activated pump. This operation relies on low friction between the rod
and the tubing wall; however, severe wear may result from sharp
bends and other directional changes in the wellbore profile (Matthews and Dunn 1993). The rod and tubing wear can be reduced
significantly by positioning rod guides at the most effective locations along the wellbore (i.e., typically, at highly tortuous sections). It is therefore crucial to identify these sections and to
analyze the wellbore geometry of such regions in detail.
Because the tortuosity-related problems in the completion or
production phases are caused by the actual geometry of the drilled
wellbore, they can be addressed properly only through case studies involving survey data from the actual well. To our knowledge,
no method for such analysis was demonstrated to date.
C 2016 Society of Petroleum Engineers
Copyright V
This paper (SPE 173103) was accepted for presentation at the SPE/IADC Drilling
Conference and Exhibition, London, 17–19 March 2015, and revised for publication. Original
manuscript received for review 21 November 2014. Revised manuscript received for review
11 June 2015. Paper peer approved 6 September 2015.
DLS. First, we need to warn against confusing the tortuosity phenomenon itself, as defined previously or by similar definitions,
with parameters used for characterization of the phenomenon.
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92
Large scale
(wellbore)
100–1000 m
Medium scale
(survey interval)
10–100 m
Small scale
1–10 m
Inclination (degrees)
HRCG
MWD
High-Resolution Survey Data. The tortuosity analysis demonstrated in this paper was applied to real continuous gyro survey
data, sampled at intervals of typically 0.3 to 1 m (1 to 3 ft). However, the methodology itself does not require any specific tool or
service. Any quality-checked survey data set with sufficiently
high resolution can be used as input, from measurement while
drilling (MWD) gyro, inertial or other tools. High-resolution data
are not usually collected in standard surveys, because they are not
2
91
90.5
90
Fig. 1—Wellbore tortuosity may occur on any length scale, here
exemplified by wellbore scale (left), survey scale (middle), and
subsurvey scale (right). Large-scale tortuosity corresponds to
planned wellbore curvature.
89.5
1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600
MD (m)
(a)
15
HRCG
MWD
DLS (degrees/30 m)
Tortuosity can be described theoretically as periodic or statistical fluctuations superimposed on a smooth wellbore trajectory
(Sugiura and Jones 2008; Samuel and Liu 2009). More important
in our context are tortuosity parameters derived from survey data.
Commonly used parameters are the DLS and its inverse, the radius of curvature (Sawaryn and Thorogood 2005). DLS is defined
as the angular change in wellbore direction over a certain MD
interval, divided by the interval (the course length). It is usually
expressed in units of (degrees/100 ft) or (degrees/30 m).
Survey-derived DLS, and more sophisticated parameters that
are derived from DLS, are widely used in the drilling-optimization computer models mentioned previously (Samuel and Liu
2009; Brands and Lowdon 2012; Mitchell et al. 2015). However,
it has been realized that dogleg parameters alone are difficult to
use as reliable tortuosity indicators (Matthews and Dunn 1993;
Oag and Williams 2000; Brands and Lowdon 2012). In particular,
Matthews and Dunn (1993) pointed out that “… dogleg severity
values provide a measure of only absolute curvature …”, implying that important information on the curvature’s orientation in
3D space is not conveyed by the dogleg parameter alone.
Another fundamental problem is the noisy character of DLS
for short course lengths. Surveyed DLS is traditionally based on
directional data at MD intervals corresponding to one drill stand
(30 m) or one individual section (10 m). However, attempts to
evaluate DLS at shorter intervals (down to 1 m or lower) show
that it becomes unstable as the interval is decreased. An example
is shown in Fig. 2. This effect was also observed by Sugiura and
Jones (2008) and by Brands and Lowdon (2012); the latter authors
compare DLS curves over course lengths of 3 and 30 m and conclude: “Although it is possible to obtain survey data in shorter
(1–3 m) intervals today with continuous MWD surveys, it is difficult for engineers to interpret the presented hole curvature (DLS).
The reported hole curvature can vary significantly on listings of
the same borehole depending on the reported survey interval.”
The noise is an intrinsic consequence of the DLS algorithm:
On the basis of the directions at the wellbore section’s endpoints
only, the direction difference may change significantly when calculated over a slightly different length, and this change is amplified when divided by a short course length. This contradicts the
behavior that should be expected by tortuosity as a physical phenomenon—namely, a smooth variation with length scale.
To overcome these limitations of DLS, this paper presents new
parameters for characterizing tortuosity—in particular, smallscale tortuosity.
91.5
10
5
0
1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600
MD (m)
(b)
Fig. 2—(a) Inclination data from a conventional MWD survey
with resolution of 10 to 30 m (red curve), and a high-resolution
continuous gyro (HRCG) survey with resolution 0.3 m (black
curve). (b) DLS on the basis of inclination and azimuth data
from the same two surveys.
needed for calculation of the wellbore position. However, high-resolution surveys with continuous MWD or continuous gyro instruments were run for various reasons, such as improved wellbore
positioning (Stockhausen and Lesso 2003), formation evaluation
(Bordakov et al. 2009), drilling-performance evaluation (Sugiura
et al. 2013), and evaluation of load-induced drillstring buckling
(Menand et al. 2006; Weltzin et al. 2009; Mitchell and Weltzin
2011). In the latter of these applications, the development of
buckling was evaluated by comparing different surveys, not by
analyzing a single survey, which is central to the methods
described here.
The high-resolution data presented in the papers mentioned
previously and in similar papers are typically limited to inclination and azimuth data, and no attempts are made to process tortuosity characteristics from these data. To the authors’ knowledge,
no comprehensive method for tortuosity analysis on the basis of
high-resolution survey data was ever published.
Theory
Preprocessing. Raw Data Set. One basic implication of the definition of tortuosity used here is that tortuosity can, in principle,
exist on any length scale. Small-scale tortuosity (on a length scale
smaller than traditional survey intervals of 10 to 30 m MD) will
be the primary subject of the analysis presented in this paper,
although the methods can be applied on any length scale, as long
as a sufficient number of data points is available.
It is assumed that the data are provided by a high-resolution
wellbore survey [i.e., at intervals of typically 0.3 to 1 m (1 to 3
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Large-scale T
Small-scale T
S
Survey
stations
L
Fitted straight line
L
Fig. 3—Establishing a local reference line for the wellbore trajectory: An analysis window W of constant MD length is moved
stepwise down the wellbore, and at each location of W, a
straight line is fitted through the position coordinates of the
survey stations covered by W. The illustration shows a wellbore
section with large-scale tortuosity on the left-hand side, and
small-scale tortuosity to the right. The well path will, in general,
be 3D, and the straight lines are fitted in 3D space.
ft)]. MD, inclination, and azimuth values are collected at these
intervals along the wellbore section of interest. It is further
assumed that the data have passed standard quality-control procedures. No correction is needed for the possible nonuniformity of
the MD data. The survey data are converted into position increments dNr,j (north), dEr,j (east), dVr,j (vertical) at each measurement station j, where the subscript r indicates that these are raw
data (i.e., input data to the analysis). Conversion to position is
performed by the tangential method (API D20 1985). This
method is not recommended for normal survey intervals (10 to
30 m) because of poor accuracy; however, for the short-interval
data, initial comparison with other methods showed that the conversion method has no significant effect on the tortuosity results.
The positions Nr,j, Er,j, Vr,j obtained by cumulating the increments are termed “the raw data set.” It is assumed that these data
constitute a true representation of the wellbore trajectory. No
other data, in particular, no knowledge of the drilling process, are
needed in the analysis.
Local Reference Line. An analysis window W is defined
along the wellbore (Fig. 3), with an MD length of typically LW 10 to 30 m (30 to 100 ft). As the window is subsequently shifted
along the wellbore, its length is kept constant, except for negligible adjustments so that the window always begins and ends on a
survey station. The window typically covers 30 to 100 stations. A
straight line is fitted through the position coordinates of these stations, by a least-squares fit in three dimensions, with equal freedom in all directions. Mathematically, this is equivalent to a
principal-component analysis (Davis 1986). The straight line will
pass through the point given by the arithmetic means N0, E0, V0 of
the coordinates:
1 XMW
N ;
N0 ¼
j¼1 r; j
MW
1 XMW
V ;
V0 ¼
j¼1 r; j
MW
1 XMW
E0 ¼
E ;
j¼1 r; j
MW
ð1Þ
where MW is the number of stations covered by the window W.
The direction of the line is given by the eigenvector that corresponds to the largest eigenvalue of the matrix
P¼
2
XMW
XMW
ðNr; j N0 Þ2
ðNr;j N0 ÞðEr;j E0 Þ
j¼1
j¼1
6X
XMW
6
MW
6
ðE
E
ÞðN
N
Þ
ðEr;j E0 Þ2
r;j
0
r;j
0
6
j¼1
4 Xj¼1
XMW
MW
ðVr;j V0 ÞðNr;j N0 Þ
ðV V0 ÞðEr;j E0 Þ
j¼1
j¼1 r;j
S
XMW
ðNr;j N0 ÞðVr;j V0 Þ
3
7
7
ðEr;j E0 ÞðVr;j V0 Þ 7
7:
5
XMW
2
ðV V0 Þ
j¼1 r;j
j¼1
XMW
j¼1
ð2Þ
The straight line is used as a reference in the further
processing.
The window W is shifted along the wellbore, and the previous
processing is performed at each location of the window. From one
location to the next, W is shifted by the average survey interval. This
implies that, typically, one survey station is discarded at one end of
the window, whereas a new station is included at the other end.
Fig. 4—Tortuosity T can be defined by the ratio between actual
(along wellbore) length S and straight-line length L. The dashed
vertical line approximately separates between a region with
large-scale tortuosity (left) and a region with small-scale tortuosity (right). The well path will, in general, be 3D, and S and L
are calculated in 3D space.
Smoothed Data Set. On the basis of the straight reference
lines produced at each location of the window W, we may derive
a new wellbore trajectory where the small-scale tortuosity is suppressed, hence, the term “smoothed data set”. The basic criterion
for one point to belong to this data set is that it lies close to the
centerpoint of the raw data subset used to establish the respective
reference line. The obvious choice is the “center of gravity” point
(N0 E0 V0) (Eq. 1) for each window location, although other definitions may be used. The set of smoothed data points will trace
out the overall (i.e., large-scale) shape of the wellbore, and this
set will be almost insensitive to single raw data outliers. By their
very nature as “outliers,” such points should be considered as constituting the small-scale tortuosity.
The length scale indicating the transition from large scale to
small scale, and the degree of suppression of small-scale tortuosity, will be closely related to the length of the W window.
Similar to the raw data set, the smoothed data set can be subject to the processing of the tortuosity parameter T (below). This
is not pursued further in this paper. The possible benefits of processing both the raw and the smoothed data sets are outlined in the
Discussion section.
Quantification of Tortuosity. Tortuosity Parameter. Tortuosity,
described as “a factor that accounts for effective elongation of fluid
paths”, has long been an important parameter in the study of flow
in porous media in petrophysics (Matyka and Koza 2012). In its
most basic form, tortuosity T was initially described by the pioneers
Kozeny and Carman (hence, the subscript KC) by the formula
TKC ¼ Spore =Lslab ; . . . . . . . . . . . . . . . . . . . . . . . . . . ð3Þ
where Spore is the distance along a flowline (following a pore)
through a porous slab of thickness Lslab. More sophisticated formulas were later developed to account for the complexities of the
flow problem. However, Eq. 3 is very appealing in the context of
this study because of the well-defined geometry of the wellbore,
and the type of data available.
It follows from Eq. 3 that TKC is always greater than or equal
to unity, and attains the minimum value of unity for a perfectly
straight path. In the wellbore situation, the analyzed section will
normally be very close to straight, and initial testing showed that
TKC from Eq. 3 typically lies in the range from 1.000 to 1.001.
Thus, to make it easier to visualize the results, we modified the
equation and define our tortuosity parameter T as
T ¼ ðS=LÞ 1
¼ ðS LÞ=L; . . . . . . . . . . . . . . . . . . . . . . . . . .
ð4Þ
where S is the along-hole length of the wellbore section being
studied and L is the straight-line distance between the ends of the
section. T is thus the relative elongation of S with respect to L.
The situation is illustrated in Fig. 4. A nice feature of the modified
definition is the favorable correspondence between the verbal
expressions “no tortuosity” or “absence of tortuosity,” meaning a
perfectly straight section, and the numeric value T ¼ 0.
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High Side Displacement (in.)
2
1.5
1
0.5
0
–0.5
–1
–1.5
–1
–0.5
0
0.5
1
1.5
Lateral Displacement (in.)
(a)
6
High Side (in.)
4
2
0
–2
6380
6360
–4
6340
–6
4
6320
2
0
–2
Lateral (in.)
(b)
–4
MD (ft)
6300
Fig. 5—Transverse-displacement vectors calculated at 1-ft
intervals over a 100-ft wellbore section, from real survey data
(cased hole). (a) Endpoints of displacement vectors plotted in
the plane transverse to the wellbore [reference line 5 z-axis,
perpendicular to the paper plane through (0,0)]. The endpoints
indicate a certain amount of wellbore spiraling. (b) The same
displacement vectors show the real shape of a wellbore or casing when applied to a 3D model. The color indicates the effective diameter value (refer to Fig. 8).
Both S and L are calculated in 3D space, and are easily determined from wellbore-position coordinates [the raw data set (Nr,j,
Er,j, Vr,j), or the smoothed data set]. The steps necessary to determine T are as follows:
1. Select S0 as the nominal length scale at which the analysis
is wanted. S0 might be the length of one particular downhole device.
2. Apply an analysis window of length S0 at a certain well
depth.
3. Adjust S0 slightly so that the window starts and ends on two
survey stations.
4. Calculate S as the sum of all intermediate survey interval
lengths between each pair of adjacent survey stations. For
raw data, this sum is equivalent to the original MD interval
between the end stations. For smoothed data, each intermediate interval must be calculated.
5. Calculate L as the straight-line distance between the start
and end stations.
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6. Calculate T according to Eq. 4.
7. Move the analysis window to another depth and repeat
Steps 2 through 6. This will produce T as a function of MD.
By changing the nominal analysis-window length S0, the tortuosity parameter can be obtained for various length scales. This, and
the option to calculate T from either raw or smoothed data, allows
for the separation of T into large-scale and small-scale tortuosity,
as suggested by Gaynor et al. (2002). This is described further in
the Discussion section.
Transverse Displacements. The transverse displacement at a
wellbore location is its deviation from a straight reference line at
that location. The processing is carried out over a section of the
wellbore at one particular depth. The along-hole length of the section and its location are determined by the user.
The initial processing step is identical to the straight-line fitting procedure described previously. An analysis window V is
used to select a subset MV of raw data points at the desired depth.
In principle, V is independent of the analysis window W used for
preprocessing of the raw data; however, V may also be chosen to
be the same processing window. A straight line is fitted through
the MV points by the principal-component-analysis method. This
straight line will be the reference for the subsequent calculation of
transverse displacements. It may further be interpreted as the
“predominant direction” of the wellbore section, and as a “lineof-sight” when considering the results.
The displacements for all MV raw data points are found by projecting each of these points perpendicularly onto the reference
line. It is a straight-forward calculation to express the projection
vectors in the 2D plane perpendicular to the line. Fig. 5a shows
an example of transverse displacements, calculated from realsurvey data.
The transverse displacements, and their distribution along the
analyzed wellbore section, directly describe the geometry of the
section. Thus, features such as the local amount of small-scale tortuosity, or the possible spiraling of the wellbore, can easily be
detected and quantified. Furthermore, the displacements can be
applied to 3D renderings of the wellbore section, giving a direct
visualization of the geometric shape. Examples are shown in Fig.
5b and in the Results section. In all 3D renderings, we have
assumed circular cross section of the wellbore/the casing, and
diameters corresponding to nominal casing inner diameter.
Effective Diameter. The tortuous buckling of a wellbore or
casing section will reduce the effective inner diameter of that section, similar to the area of free sight that can be seen through a
curved tunnel. The effective diameter is defined as the maximum
diameter of a straight cylinder that can be inserted into the section
without distorting it (i.e., just barely touching the inner walls). To
yield a meaningful definition, the cylinder must have a length equal
to or greater than the length of the wellbore or casing section. If the
cylinder represents a physical downhole device, the processing
window V should be chosen equal to the length of the device.
As a result of irregular borehole washouts or drillstring or
BHA wear on the formation, the openhole wellbore may not have
a circular cross section. A deviation from circular cross section
may also be found on tubings and casings for various other reasons. Such effects are neglected here, and it is assumed that the
cross sections are perfectly circular.
The reduction in diameter does not depend on the tortuosity of
the wellbore or casing section only. The diameter reduction will,
in general, also change with the orientation of the inserted cylinder. Fortunately, the cases that would occur most frequently, and
also would need the most precise analysis, are expected to be
nearly straight wellbore sections. In such cases, it is reasonable to
let the cylinder’s axis (the line of sight) coincide with the reference line established for processing of transverse displacements.
This implies that one can use the displacements directly in the calculation of the diameter reduction.
Heavily bent sections can more easily be identified and judged
as problematic, without extensive analysis. However, the processing of a reference line and transverse displacements may also be
applied to these cases. Thus, a single formula for diameter
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80
55
75
50
70
45
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40
60
35
55
30
2500
2600 2700
2800
2900
3000
3100
3200
Inclination (degrees)
Azimuth (degrees)
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3300
MD (m)
(a)
Azimuth (degrees)
55
50
45
40
HRCG
MWD
35
2500
2600
2700
2800
2900
3000
3100
3200
MD (m)
(b)
25
HRCG
MWD
DLS (degrees/30 m)
20
15
10
5
0
2500
2600
2700
2800
2900
3000
3100
3200
MD (m)
(c)
Fig. 6—Field Case 1. (a) Inclination and azimuth from conventional MWD survey. (b) Comparison of azimuth from the conventional MWD survey and from an HRCG survey. (c) DLS curves
calculated from the conventional MWD survey (inclination and
azimuth), and from the HRCG survey (inclination and azimuth).
reductions can be developed, which is valid for both low- and
high-tortuosity regions. With the assumption of a circular cross
section and a sufficiently high number MV of displacements (the
number is discussed in the Discussion section), a good estimate
for the effective diameter Deff is obtained from:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxi xj Þ2 þ ðyi yj Þ2 ; when
Deff ¼ Dnom maxi;j
;
Dnom maxi;j ð…Þ
Deff ¼ 0;
when Dnom < maxi;j ð…Þ ð5Þ
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where xi (xj) and yi (yj) are the displacement coordinates in the
transverse plot shown in Fig. 5a, and maxi,j(…) means the maximum of the square-root function encountered over all possible
combinations of i ¼ 1…MV, and j ¼ 1…MV. Dnom is the nominal
inner diameter of the wellbore or casing. If the true diameter is
measured, for example, by a caliper, these values might be used
for Dnom; however, a diameter varying with MD or a noncircular
cross section would require a more sophisticated equation for
Deff. We did not pursue this further in this study; thus, all processed examples assume a nominal and constant value for Dnom.
Deff is calculated with the length of the analysis window,
which represents the downhole device, as the only external parameter. By shifting the analysis window throughout the well and
calculating the effective diameter at each location, we obtain a
graph of Deff vs. MD, for the selected device length. This graph
may be plotted directly, or the results may be superimposed on a
3D visualization of the wellbore or casing shape. Examples are
given in the following section.
Results
In all field cases, the tortuosity was analyzed from a high-resolution continuous gyro (HRCG) survey, which was run in cased
hole (post-drilling). The continuous gyro survey is sampled at
22 Hz and preprocessed to produce a survey point every 0.4 seconds. The typical running speed in casing is 0.8 m/s, yielding
0.3 m (1 ft) between measurements. Smaller survey intervals are
possible by increasing the preprocessing output rate, or by
decreasing the running speed. The survey data are qualitychecked according to procedures that are based on published error
models (Ekseth et al. 2011).
Field Case 1. In a deepwater offshore well drilled in Latin America
with a modern rotary-steerable system, a conventional MWD survey
showed a straight tangent section with small doglegs (Fig. 6), but
there were problems associated with running casing all the way to
total depth (TD) (3290 m MD).
There were indications of issues with the well on the trip out
of the hole with the 121/4-in. BHA. Back reaming was required
from TD to 2775 m MD. When running the 95/8-in. casing, the
string hung up at 3264 m MD, and it could not go any further.
Many attempts were made to pull the string free, but it remained
stuck at 3261 m MD. A cleanout run of the 95/8-in. casing was
required and resulted in abandoning several meters of casing.
Even though there were several problems in this part of the
well, the low DLS suggested that the completion equipment could
be installed here. However, two production pumps failed prematurely, and it was decided to run an HRCG survey. Comparison
between the MWD and the HRCG showed no evidence of gross
error in the MWD data but indicated rapid azimuth fluctuations
not detected by the conventional MWD survey (Fig. 6b). The
DLS, based on the HRCG data, also indicated higher tortuosity
below 3000 m MD but failed to provide a quantifiable measure
(Fig. 6c). The tortuosity parameters calculated from the HRCG
revealed much-more detail (Figs. 7 and 8) and explained very
well all the issues with the casing and the production equipment.
The evidence from the HRCG resulted in a decision to place
the production pumps 150 meters higher than initially suggested
to avoid future rework. The initial lack of information between
the conventional MWD survey stations (30-m separation) resulted
in several weeks of lost production and significant extra costs for
the operator (estimated to be more than USD 7 million).
Field Case 2. A land well at high-latitude north was drilled and
surveyed with conventional MWD systems. The well profile is
indicated in Fig. 9a. On the basis of the DLS plots (Fig. 9b), completion equipment was placed near 4,500 ft MD.
After only a few months of operation, the production pump
failed, and an HRCG survey was requested to evaluate the geometry and tortuosity of the wellbore. As in the previous example, no
gross error was identified in the conventional MWD survey, and
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2.5
Tortuosity (T )
2
1.5
1
0
2700
2800
2900
MD (m)
3000
3100
3200
3000
3100
3200
Maximum Outer Diameter of Device (in.)
(a)
8
7
6
5
4
3
2
1
0
2500
2600
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Field Case 3. A land well in the southern USA was surveyed
with conventional MWD methods. The profile is shown in
Fig. 12a. The well is vertical down to 6,300 ft MD and then builds
rapidly to horizontal, reaching 90-degrees inclination at approximately 7,500 ft MD and remaining close to horizontal thereafter.
To achieve maximum drawdown, it was required to position an
electrical submersible pump (ESP) in the near horizontal-well section at the maximum true vertical depth at which pump placement
was physically possible.
To achieve the required pumping capacity, it was proposed to
deploy an ESP of length 124 ft and diameter 4 in. in the well. The
planned tangent section for pump placement was 7,626 to 7,934 ft
MD. On the basis of the conventional MWD survey and DLS
(Fig. 12b), the pump could have been placed at any depth between
7,750 and 7,950 ft MD.
An HRCG survey was requested to confirm the planned pump
location. The tortuosity analysis leads to a very different conclusion (Fig. 12c). The effective diameter curve indicates that the
depths at which an ESP may be safely deployed in this well are
far more restricted than is evident from the DLS analysis. The
best locations at which a pump may be deployed are shown to be
at depths of 7,880 ft MD or possibly at 7,815 ft MD (depths refer
to center of pump), whereas depths less than 7,800 ft MD and
greater than 7,890 ft MD should be avoided altogether. The Deff
curve indicates that, if positioned at 7,880 ft MD, the ESP may
have a maximum diameter of 3.1 in. to avoid contact with the casing, unless its length is shortened. This information led to a reconsideration of the installment plans.
0.5
2600
Page: 6
the coarse resolution of the conventional MWD data (100 ft) led
to the initial conclusion that 4,400 to 4,500 ft MD was the region
with lowest tortuosity. The HRCG data (interval ¼ 1 ft) indicated
that the low-tortuosity region actually is found at 4,300 to 4,400 ft
MD. This is confirmed by the effective-diameter plot (Fig. 10b),
which further shows that, at 4,500 ft MD, there is no free line of
sight over a length of 100 ft, and therefore this location should be
avoided. A new pump was placed near 4,300 ft MD on the basis
of this information. Fig. 11 shows the actual shape of the cased
wellbore at the two locations.
×10–4
2500
Stage:
2700
2800
2900
MD (m)
(b)
Fig. 7—Field Case 1. Tortuosity curve T (a) and effective diameter curve Deff (b) calculated from HRCG survey raw data with
processing window lengths S0 5 30 m and V 5 30 m, respectively. The nominal casing inner diameter is Dnom 5 8.83 in.
Discussion
Processing and Results. Both the tortuosity parameter T and the
transverse displacements represent geometric features of the wellbore trajectory that can be intuitively understood—relative elongation of the trajectory compared with the straight line, and
when the final positioning of the well was evaluated, the difference in coordinates between the two surveys was only 10 ft.
A good overall correspondence can be observed between the
DLS curve (Fig. 9b) and the tortuosity plot (Fig. 10a). However,
8
2500
2550
7
2600
2650
6
2700
1900
Vertical (m)
2750
2000
5
2800
2850
2900
2100
4
2950
2200
2300
1000
3
3000
700
3050
3100
3150
1100
900
1200
1000
1300
North (m)
1400
2
800
1
East (m)
0
1100
1500
Fig. 8—Field Case 1. Wellbore trajectory with superimposed transverse displacements over the whole HRCG survey interval 2500
to 3200 m MD. The analyzing window length is V 5 30 m. Transverse dimensions (wellbore diameter and displacements) were augmented by a factor of 100 for clarity. The Deff curve (Fig. 7b) was added in terms of color coding, according to the color bar shown
on the right (blue 5 no diameter reduction from nominal 8.83 in.; red 5 diameter reduced to zero).
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75
Page: 7
Total Pages: 10
×10–4
75
45
60
30
45
15
30
0
15
4.5
Tortuosity (T )
60
Inclination (degrees)
Azimuth (degrees)
5
4
3.5
3
2.5
2
1.5
1
MD (ft)
(a)
6
DLS (degrees/100 ft)
5
4
3
2
1
0
0
0
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000
500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000
MD (ft)
(b)
Fig. 9—Field Case 2. Inclination and azimuth (a) and DLS (b)
from conventional MWD survey (interval 100 ft).
sideways deviations from the straight line. The algorithms apply
data from a single survey and just one input parameter—length of
the analysis window, which effectively defines the length scale
of the tortuosity results. These features make the methodology
simple, robust, and superior to established methods such as the
DLS analysis.
The accuracy of the results is considered to be good. Using all
information within the analysis window (from typically 30 stations), the parameters are virtually insensitive to small changes in
the number of stations as the window is moved along the wellbore.
Such small changes may occur because of the possible nonuniformity of the survey intervals. In particular, the ratio S/L in the T
parameter will be even less sensitive to these changes than S or L
alone. Furthermore, the use of all information within the window
implies that the parameters increase or decrease smoothly if the
window length is gradually changed. In early tests for both T and
Deff, the results showed this smooth behavior, indicating reliable
algorithm outputs, as long as the number of stations was typically
>10. For survey report intervals of 0.3 m, for example, tortuosity
can therefore be analyzed on length scales down to 3 m. The resolution interval of the output, on the other hand, is much smaller
than the window length. According to the Shannon-Nyquist theorem (Ifeachor and Jervis 1993), the highest possible resolution is
twice the sampling interval (theoretical limit, for band-limited signals); this amounts to 0.6 m for the previous example.
The tortuosity analysis can be applied to data from openhole
surveys, or from surveys inside drillstring, casing, or production
tubing. The results should be expected to differ among all these
cases because of possible buckling and misalignment of drillstring
or tubing inside the hole/casing, and changes to borehole-wall
roughness and undulations caused by insertion of the casing. One
application of the methodology may therefore be to analyze such
500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000
MD (ft)
(a)
Maximum Outer Diameter of Device (in.)
–15
0
0.5
9
8
7
6
5
4
3
2
1
0
0
500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000
MD (ft)
(b)
Fig. 10—Field Case 2. Tortuosity curve T (a) and effective-diameter curve Deff (b) on the basis of HRCG data and a device
length of 100 ft. The nominal casing inner diameter is
Dnom 5 8.83 in.
differences; however, this is outside the scope of this study. In all
field cases presented here, the HRCG survey was run in cased hole.
For cases such as Field Case 3, the feasibility of other device
dimensions can be evaluated easily by changing the length of the
Deff analysis window. However, this was not performed in the
present case. Furthermore, in situations where the tortuosity leads
to bending of equipment, it would be possible to evaluate bending
moments and final equipment shape by combining our detailed
tortuosity description with the geometry and the material (stiffness) parameters of the device.
It should also be emphasized that the methodology proposed
in this paper relates only to the geometric shape of the wellbore,
and factors such as reservoir pressure, tubing stiffness, or dynamic
processes are not considered. Thus, the results should be interpreted as recommendations that are based on wellbore shape only.
Factors other than tortuosity may be considered more important,
and may therefore become the major factors in decisions regarding equipment installation and operation.
Further Work. The smoothed data set proposed in the Theory
section has not been examined in detail in this study. It is believed
that it can be used to separate between large-scale and small-scale
tortuosity, along the lines suggested by Gaynor et al. (2002). The
reason for this is that tortuosity details can be processed with a
maximum resolution equaling approximately twice the survey
interval from the raw data set, whereas the smoothing procedure
will efficiently suppress tortuosity on length scales shorter than
the analysis window W, which typically is 10 to 30 times the survey interval.
As mentioned in the Theory section, Dnom in Eq. 5 is assumed
to be the nominal inner diameter of a wellbore or casing with a
constant and circular cross section. On the basis of the field
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High Side (in.)
10
5
0
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400
90
350
75
300
60
250
45
200
30
150
15
4,550
–5
–10
10
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Azimuth (degrees)
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4,500
5
100
0
6,000 6,200 6,400 6,600 6,800 7,000 7,200 7,400 7,600 7,800 8,000
MD (ft)
MD (ft)
0
–5
4,450
Lateral (in.) –10
(a)
(a)
DLS (degrees/100 ft)
6
8
4
2
0
4
3
2
1
–2
–4
7,550 7,600 7,650 7,700 7,750 7,800 7,850 7,900 7,950 8,000
MD (ft)
4,350
–6
6
4
2
0
–2
Lateral (in.)
–4 4,250
(b)
4,300
MD (ft)
(b)
Maximum Outer Diameter of Device (in.)
High Side (in.)
6
5
4
3.5
Fig. 11—Field Case 2. Wellbore shape computed from transverse displacements over 100-ft sections at two depths: (a)
4,500 ft MD (high-tortuosity region with Deff 5 0 in.) and (b)
4,300 ft MD (low-tortuosity region with Deff 6.7 in.). The nominal casing inner diameter is Dnom 5 8.83 in., and the colors indicate the calculated Deff values (refer to Fig. 8).
examples, we believe that this gives sufficient accuracy for the
present application. However, more accurate estimates for Deff
would be obtained when using real Dnom values, for example, from
caliper measurements. This also may require a more sophisticated
formula for Deff. Furthermore, the Deff analysis can be extended for
devices with noncircular or nonuniform (along the device) cross
sections, and eventually allowing for some bending of the device,
within the manufacturer’s or operator’s specifications.
The T and Deff parameters may respond differently to various
tortuosity geometries, as illustrated in Table 1. Therefore, by combining the information from the two parameters, we may obtain
more detailed information on the nature of the tortuosity. For
Wellbore Shape (local)
3
2.5
2
1.5
1
0.5
0
7,500 7,550 7,600 7,650 7,700 7,750 7,800 7,850 7,900 7,950 8,000
MD (ft)
(c)
Fig. 12—Field Case 3. (a) Inclination and azimuth from conventional MWD survey. (b) DLS calculated from the conventional
MWD survey. (c) Effective diameter Deff curve on the basis of
HRCG data and a device length of 124 ft. The nominal casing
inner diameter is Dnom 5 6.28 in.
Relative Elongation, T = (S–L)/L
Span of Displacements
Effective Diameter, Deff
0
0
Dnom (=max)
Small
Small
Near Dnom
Large
Large
Small
Large
Small
Near Dnom
Table 1—Response of tortuosity parameters T and Deff to various local shapes of the wellbore. The wellbore trajectories may be 3D.
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example, if T is large, as shown in the lower two lines of the table,
the value of Deff may help to determine whether the elongation is
caused by a single “long” curve (still within the analysis window),
or caused by several microbends. A “shape parameter” derived
from T and Deff, or from T and the span of transverse displacements,
could be presented as a graph vs. MD, which may be more convenient than 3D renderings over large wellbore sections.
Conclusions
This paper has presented a novel methodology to analyze wellbore tortuosity, in particular, tortuosity on a length scale shorter
than conventional survey report intervals (30 m). The input data
are provided by a single high-resolution directional survey, which
may use any surveying tool or procedure capable of delivering
such high-resolution data. By applying a user-defined data window as a single external parameter, the method quantifies tortuosity on any length scale greater than approximately 10 times the
survey-data interval, and with a maximum resolution that equals
twice the data interval, in terms of
• Tortuosity parameter; the relative elongation of along-hole
length with respect to straight-line length.
• Transverse displacements of the wellbore trajectory from a
locally established reference line.
• Effective inner diameter of wellbore or casing as a result of
small-scale bending.
The outputs can be presented individually or in combination,
either as graphs vs. MD or in 3D renderings showing the actual
geometry of the wellbore. The results are easily visualized, interpreted, and communicated to operating personnel, and represent a
significant improvement from traditional tortuosity analysis methods on the basis of DLS.
The methodology was applied to various field cases in which
the results demonstrate the identification of highly tortuous
regions that were unnoticed or poorly quantified by traditional
methods. This information has direct relevance to decisions
regarding the installation of equipment in the completion and production phases.
The tortuosity analysis may thus contribute to more efficient
operations and improvements to equipment functionality, eventually leading to reduced workover frequency and considerable cost
savings. The possible applications of the methodology range from
evaluation of the drilling process to the installation of completionand-production equipment.
Nomenclature
dNr,j, dEr,j, dVr,j ¼ raw data position increments in N, E, V at station j, ft or m
Dnom ¼ nominal inner diameter of casing or wellbore,
in.
Deff ¼ effective diameter; maximum-allowed outer
diameter of downhole device, in.
L ¼ straight-line distance (between start and end
stations of a section), ft or m
MV ¼ number of survey stations covered by V window
MW ¼ number of survey stations covered by W
window
N, E, V ¼ position coordinates: Northing, Easting, Vertical depth, ft or m
N0, E0, V0 ¼ average (arithmetic mean) of N, E, V over a
subset of survey stations, ft or m
Nr,j, Er,j, Vr,j ¼ raw data position coordinates N, E, V of station j, ft or m
P ¼ matrix used in principal-component analysis
S ¼ along-hole distance; actual value of S0, ft or m
S0 ¼ analysis window for tortuosity parameter;
length of this window, ft or m
T ¼ tortuosity parameter (Eq. 4)
TKC ¼ tortuosity parameter in petrophysics (Eq. 3)
V ¼ analysis window for effective-diameter
analysis
Stage:
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W ¼ analysis window for establishing reference
line
xi, yi, xj, yj ¼ coordinates of displacement vectors i and j in
plane transverse to reference line, in.
Acknowledgments
The authors want to thank Gyrodata and SandRidge Energy for
permission to publish this work, and for providing field data. Furthermore, Rob Shoup, Gyrodata, is thanked for stimulating
discussions.
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Jon Bang has been a development engineer at Gyrodata
since 2013. Previously, he was with SINTEF Petroleum Research,
Norway, working on wellbore-position uncertainty analysis.
Bang’s current research interests include error modeling, quality control, and data processing related to wellbore surveying.
He holds an MSc degree in electronics and a PhD degree in
laser physics, both from the Norwegian University of Science
and Technology. Bang is a member of SPE, and an active
member of the Industry Steering Committee on Wellbore Survey Accuracy (ISCWSA).
Onyemelem Jegbefume is a research and development engineer at Gyrodata. Previously, he worked for 3 years as a systems
engineer for Rockwell Collins. Jegbefume’s current research
interests include algorithms for wellbore surveying, communication systems, and signal processing for oilfield applications.
He holds a PhD degree in electrical engineering from the University of Texas at Dallas. Jegbefume is a member of SPE.
Adrián Ledroz is the Vice President, Survey Technologies, Technical Services for Gyrodata. He has worked with gyroscopes
for the last 15 years, and his research interests include inertial
sensors, data processing, quality control, and error modeling
related to wellbore surveying. Ledroz holds a BSc degree in
biomedical engineering from Universidad Nacional de Entre
Rı́os, Argentina, and an MSc degree in electrical engineering
from the University of Calgary, Canada. He is a member of
SPE, and an active member of ISCWSA.
John Weston is the Global Adviser Gyro Technology for Gyrodata, and his research interests are in inertial systems and their
application to wellbore surveying. He holds a BSc degree in
electrical engineering from the University of Wales and an
MSc degree in information and systems engineering from the
University of Birmingham, UK. Weston is coauthor of the textbook Strapdown Inertial Navigation Technology. He is a member of the Institution of Engineering Technology and SPE, and
is an active member of the ISCWSA.
John H. (Jay) Thompson is the Artificial Lift Specialist with SandRidge Energy in Oklahoma City, Oklahoma. He has been
with SandRidge for 3 years but has been in the pumping industry, both surface pumping and downhole, for more than 26
years. Thompson’s interests include the effects of wellbore
design and how it affects ESP pumps, root-cause failure analysis, and how the longevity of downhole equipment can be
improved. He is a member of SPE.
2016 SPE Production & Operations
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