BACKGROUND
Directional drilling is the process of directing the borehole along a defined trajectory.
Deviation control during drilling is the process of keeping the borehole trajectory contained
within specified limits, e.g., limits on the inclination angle or distance from the defined
trajectory, or both. Both have become important to developers of unconventional hydrocarbon
resources.
Various drill string steering mechanisms exist to provide directional drilling and
deviation control: whipstocks, mud motors with bent-housings, jetting bits, adjustable gauge
stabilizers, and the increasingly popular rotary steering systems (RSS). These techniques each
employ side force, bit tilt angle, or some combination thereof, to steer the drill string' s forward
and rotary motion. However, the resulting borehole's actual curvature is not determined by
these parameters alone, and it is generally difficult to predict, necessitating slow drilling and
frequent survey measurements.
Among the most important trajectory parameters that need to be monitored and
controlled is the wellbore dogleg, i.e., the rate at which the trajectory changes direction. The
rate of such direction changes can be expressed in terms of degrees per unit length or
alternatively in terms of the radius of curvature. Decreasing the curvature radius corresponds
to increasing the degrees of directional change per unit length, both of which correspond to
increasing the dogleg severity. Severe doglegs create a number of difficulties including casing
insertion difficulty, increased friction, increased casing wear, and increased likelihood of
bottomhole component trapping.
One method for measuring borehole curvature and, more specifically, dogleg severity,
is to measure the bending of a bottomhole assembly as it passes along the borehole. A subtle
yet important shortcoming of this method arises from the erroneous assumption that the
bottomhole assembly bends in the same fashion as the borehole.
BRIEF DESCRIPTION OF THE DRAWINGS
Accordingly, there are disclosed herein systems and methods that employ improved
estimation of wellbore dogleg from tool bending moment measurements. In the drawings:
Fig. 1 is a schematic diagram of an illustrative well drilling environment.
Fig. 2 is a function-block diagram of a logging while drilling (LWD) system.
Figs. 3a and 3b are borehole cross-sections with drillstring trajectory deviations.
Fig. 4 is a flow diagram of an illustrative wellbore dogleg estimation method.
Fig. 5 is a force diagram for the end nodes of a drill string segment.
Fig. 6 is a diagram showing local curvature radii for drill string segments on each side
of a given node.
Fig. 7 is a diagram showing trajectory interpolation between nodes.
It should be understood, however, that the specific embodiments given in the drawings and
detailed description thereto do not limit the disclosure. On the contrary, they provide the
foundation for one of ordinary skill to discern the alternative forms, equivalents, and
modifications that are encompassed together with one or more of the given embodiments in
the scope of the appended claims.
DETAILED DESCRIPTION
To provide context and facilitate understanding of the present disclosure, Fig. 1 shows
an illustrative drilling environment, in which a drilling platform 102 supports a derrick 104
having a traveling block 106 for raising and lowering a drill string 108. A top-drive motor 110
supports and turns the drill string 108 as it is lowered into the borehole 112. The drill string's
rotation, alone or in combination with the operation of a downhole motor, drives the drill bit 114
to extend the borehole. The drill bit 114 is one component of a bottomhole assembly (BHA) 116
that may further include a rotary steering system (RSS) 118 and stabilizer 120 (or some other
form of steering assembly) along with drill collars and logging instruments. A pump 122
circulates drilling fluid through a feed pipe to the top drive 110, downhole through the interior
of drill string 8, through orifices in the drill bit 114, back to the surface via the annulus around
the drill string 108, and into a retention pit 124. The drilling fluid transports cuttings from the
borehole 112 into the retention pit 124 and aids in maintaining the integrity of the borehole. An
upper portion of the borehole 112 is stabilized with a casing string 113 and the lower portion
being drilled is open (uncased) borehole.
The drill collars in the BHA 116 are typically thick- walled steel pipe sections that provide
weight and rigidity for the drilling process. The thick walls are also convenient sites for installing
logging instruments that measure downhole conditions, various drilling parameters, and
characteristics of the formations penetrated by the borehole. Among the typically monitored
drilling parameters are measurements of weight, vibration (acceleration), torque, and bending
moments at the bit and at other selected locations along the BHA. The BHA 116 typically further
includes a navigation tool having instruments for measuring tool orientation (e.g., multicomponent
magnetometers and accelerometers) and a control sub with a telemetry transmitter
and receiver. The control sub coordinates the operation of the various logging instruments,
steering mechanisms, and drilling motors, in accordance with commands received from the
surface, and provides a stream of telemetry data to the surface as needed to communicate relevant
measurements and status information. A corresponding telemetry receiver and transmitter is
located on or near the drilling platform 102 to complete the telemetry link. The most popular
telemetry link is based on modulating the flow of drilling fluid to create pressure pulses that
propagate along the drill string ("mud-pulse telemetry or MPT"), but other known telemetry
techniques are suitable. Much of the data obtained by the control sub may be stored in memory
for later retrieval, e.g., when the BHA 116 physically returns to the surface.
A surface interface 126 serves as a hub for communicating via the telemetry link and for
communicating with the various sensors and control mechanisms on the platform 102. A data
processing unit (shown in Fig. l a as a tablet computer 128) communicates with the surface
interface 126 via a wired or wireless link 130, collecting and processing measurement data to
generate logs and other visual representations of the acquired data and the derived models to
facilitate analysis by a user. The data processing unit may take many suitable forms, including
one or more of: an embedded processor, a desktop computer, a laptop computer, a central
processing facility, and a virtual computer in the cloud. In each case, software on a non-transitory
information storage medium may configure the processing unit to carry out the desired
processing, modeling, and display generation.
Among the various types of measurement data that may be acquired by the BHA 116 are
multi-component measurements of the earth's magnetic field and gravitational field at each of a
series of survey points (or "stations") along the length of the borehole. The survey points are
typically those positions where the navigation tool is at rest, e.g., where drilling has been halted
to add lengths of drill pipe to the drill string. The gravitational and magnetic field measurements
reveal the slope ("inclination") and compass direction ("azimuth") of the borehole at each survey
point. When combined with the length of the borehole between survey points (as measureable
from the length added to the drill string), these measurements enable the location of each survey
point to be determined using known techniques such as, e.g., the tangential method, the balanced
tangential method, the equal angle method, the cylindrical radius of curvature method, or the
minimum radius of curvature method, to model intermediate trajectories between survey points.
When combined together, these intermediate trajectories form an overall borehole trajectory.
Also among the various types of measurement data that may be acquired by the BHA
116 are caliper measurements, i.e., measurements of the borehole's diameter, optionally
including the borehole's cross-sectional shape and orientation, as a function of position along the
borehole.
Fig. 2 is a function-block diagram of an illustrative directional drilling system, though
the illustrated modules are also largely representative of a wireline logging system. One or
more downhole tool controllers 202 collect measurements from a set of downhole sensors 204,
preferably but not necessarily including navigational sensors, drilling parameter sensors, and
formation parameter sensors, to be digitized and stored, with optional downhole processing to
compress the data, improve the signal to noise ratio, and/or to derive parameters of interest
from the measurements.
A telemetry system 208 conveys at least some of the measurements or derived
parameters to a processing system 210 at the surface, the uphole system 210 collecting,
recording, and processing measurements from sensors 212 on and around the rig in addition to
the telemetry information from downhole. Processing system 210 generates a display on
interactive user interface 214 of the relevant information, e.g., measurement logs, borehole
trajectory, drill string trajectory, or recommended drilling parameters to optimize a trajectory
to limit estimated dogleg severity. The processing system 210 may further accept user inputs
and commands and operate in response to such inputs to, e.g., transmit commands and
configuration information via telemetry system 208 to the tool controllers 202. Such commands
may alter the settings of the steering mechanism 206.
The software that executes on processing units 128 and/or 210, includes borehole
trajectory estimation program with a drill string trajectory determination module. As illustrated
by Figs. 3a-3b, the two trajectories may be quite different. For example, in Fig. 3a, the drill
string 304 follows a straighter trajectory than the borehole 302, while in Fig. 3b, the drill string
304 follows a significantly more convoluted trajectory than the borehole 302. And while the
difference in radii constrains the drill string trajectory relative to the borehole trajectory, the
two need not parallel each other or resemble each other on aught but the large scale.
Nevertheless, for a given borehole trajectory, the drill string trajectory can be estimated using
a stiff-string model or a finite element model, each of which would account for the weight
(density) of the drill string, the stiffness of the drill string, and the external forces on the drill
string.
As indicated by the illustrative method in Fig. 4, the borehole trajectory can be
determined in the following way. As drilling and/or tripping operations are ongoing in block
402, the downhole logging tools collect navigation and (optionally) borehole geometry
measurements as a function of BHA position and communicate them to the processing system.
Also collected are drilling parameters as a function of BHA position, specifically including
multi-component forces (or deformations) and bending moments at the bit and at selected
positions along the BHA. Strain gauges may be used to provide the deformation and bending
moment measurements. Multi-component forces can be measured directly using multicomponent
accelerometers or indirectly derived from the strain gauge measurements. Some
contemplated embodiments further collect such measurements at distributed points along the
drill string.
In block 406, the processing system models the borehole trajectory based on the
navigation and borehole geometry measurements. This model may initially take the form of a
short, straight borehole segment. As the drilling progresses, the length and shape of the model
borehole iteratively gets updated, with the updated model from a previous time step being taken
as the initial model for the current time step.
In block 408, the processing system employs a three-dimensional finite element
analysis (3D FEA) or a stiff string analysis to derive a predicted drill string trajectory from the
current borehole model, finding a drill string and BHA curvature that accounts for the boundary
conditions and material properties of the drill string. The derived trajectory is used for
calculating the expected multi-component deformations and bending moments. In block 410,
the expected deformations and bending moments are compared to the measured deformations
and bending moments from block 404. If they do not match, the system updates the borehole
model in block 412, adjusting the borehole curvature and length in a fashion that reduces a
mismatch error between the expected and measured bending moments and deformations for
the current BHA position and all previous BHA positions.
Blocks 408-412 are repeated until the mismatch error falls below a threshold. Next, the
system determines whether the drilling/tripping operations are still going. If so, blocks 404-
414 are repeated to obtain and apply the new measurements. Otherwise, the method completes,
yielding a robust estimation of the borehole trajectory that does not presume any equivalence
with the drill string trajectory. Moreover, the estimated borehole trajectory takes into account
the boundary conditions of the drill string.
We now turn to a more detailed discussion of one approach for deriving a drill string
trajectory and updating a borehole path. It employs Finite Element Analysis (FEA), a
mathematical method of solving a complex problem by breaking it into several smaller
problems. Each of the smaller problems is then solved and their solutions combined to solve
the complex problem. The following are taken into account as part of the problem formulation:
(1) Tubular stiffness in bending; (2) Tubular joint to hole wall clearance; (3) Stiffness modified
for compressive force; (4) Single point weight concentrations. The derived drill string
trajectory yields the following: (1) Side forces, including drag and torque; (2) Bending stresses;
(3) Pipe position in the hole.
The FEA may be performed as a real-time calculation that is updated as measurements
are acquired, or as a stand-alone calculation that is performed after measurement collection is
complete. In either case, hole curvature can be modeled in reverse, i.e., by casting the situation
as one having a curved pipe inside a straight borehole. Internal moments and forces are applied
which would be required to bend the segments between adjacent nodes in the required
curvature but in the opposite direction. In this way the curvature may be handled consistently
throughout the sections being analyzed. It does not suffer from the frame-of-reference
distortion that is often problematic with finite element solutions. This approach also enables
the stiff string model to manage large deflections. This curvature formulation applies equally
in both build (inclination change) and walk (azimuth change).
Fig. 5 illustrates how the internal forces and bending moments may be defined for the
nodes at each segment along the drill string. At node n, we have force components FX(n), F (n),
Fz(n), and bending moments MX(n), M (n), MZ(n), respectively along the x, y, and z axes. (The
axial force may also be represented as Pn, which is positive whenever the axial force is
compressive.) These can be iteratively derived from the drill string trajectory that has been
derived from an estimated borehole trajectory. The derived drill string trajectory provides a
local curvature radius Rn at each segment, as indicated in Fig. 6 . The segments are stiff "beams"
of length Ln, horizontally separating end nodes that are vertically displaced by distances Dn.
The beams have a curvature stiffness of Kn and a stiffness factor S. The beams are presumed
to have a constant weight per length W that translates into a fixed end moment (FEM) Ms. The
inclination angle at each node is represented by qh (when relative to horizontal) or An (when
relative to a baseline between nodes).
The curved trajectory between nodes can be interpolated as a cubic spline as indicated
in Fig. 7 . The resulting angle, displacement, and curvature along the baseline between the nodes
is:
Ax = A+ 4.FEM .(x -3x 2 + 2x3) + A +3(A +A2).x 2 - 2(2A +A ).x
Dx = Dl +A.x.Ln_ .Ln.FEMn.(x -2x 3 +x )/SK n-Ln.(A x +(Al +A ).x -(2A l +A ).x )
Cx =— =(4.FEMn.(\ -6x +6x )/SK n +6.(A +A2).x -2.(2A +A2))/ L SL n
where, with reference to Fig. 7, we have (with the small angle approximation):
A = (Dn+1-D„)/L„
The foregoing equations account for both causes for the curvature of the string: (1) the cubic
spline that join nodes n and n+1; and (2) the weight distribution in the drill string causing a
natural sag in the middle. In the case of a string that has no external upset (tool joint), the
effect of weight may be ignored.
To determine the transferred force and moment from the previous node, the following
equations may be used (with reference to Fig. 6):
M l = SKn-l. Cn-1. qi - Tn - 1 . D l
where Tn and Sn are calculated as below
S = (P - 2T )/L
T
n-1 = SK
n-l ( + C„_i ) L
T = SK .(l + C )/L
with Cn being the carry-over factor.
Similarly, the transferred force and moment from next node can be calculated:
These transferred curvature moments and forces are iteratively added to the internal forces
and fixed end moments from weight:
until convergence is reached for each of the nodes along the drill string.
At least some embodiments of the drill string trajectory determination module employ
the nominal outer diameter of the drill string and the nominal borehole diameter, together with
stiffness of the drill string and the borehole trajectory as estimated from previous iterations or
in combination with other sources of trajectory information such as location information of
survey stations along the borehole. Alternatively, or in addition, the drill string trajectory
determination module may account for the location and size of the threaded couplings between
tubulars and the location and size of any centralizers. The drill string trajectory determination
module may still further account for forces on the drill string during the drilling process
including gravity, buoyancy, and compression. Those of ordinary skill in the art are familiar
with the use of stiff-string models and further details may be found in references such as A.
McSpadden and K.Newman, "Development of a Stiff-String Forces Model for Coiled Tubing,"
SPE-74831-MS (2002), and L. Gorokhova, A. Parry, andN. Flamant, "Comparing Soft-String
and Stiff-String Methods used to Compute Casing Centralization," SPE-163424-PA (2014).
The foregoing disclosure enables the estimation of the wellbore dogleg from the
bending moment measurements from string in both drilling and real time environment. It
enables better prediction for the performance of various drill ahead options and the
performance of various sensitivity analyses with RSS (rotary steerable systems), mud motor,
and other drill string, BHA, and bit configurations. It further enables improved borehole
trajectory information for use in modeling other drilling environment parameters, including the
prediction or calculation of mechanical, hydraulic and pneumatic properties.
Accordingly, the embodiments disclosed herein include:
Embodiment A:Aborehole curvature logging system that includes: a drill string having
a bottomhole assembly (BHA) with sensors providing actual deformation and bending moment
measurements as a function of BHA position at spaced-apart intervals on the BHA; a
processing system that retrieves said actual measurements and responsively generates a log of
borehole curvature; and a user interface that displays the borehole curvature log. The
processing system implements a method that generates the log by: providing an estimated
borehole trajectory; deriving predicted deformation and bending moment measurements based
on the estimated borehole trajectory; determining an error between the predicted measurements
and the actual measurements; updating the estimated borehole trajectory to reduce the error;
repeating said deriving, determining, and updating to refine the estimated borehole trajectory;
and converting the estimated borehole trajectory into a borehole curvature log.
Embodiment B : A borehole curvature logging method that comprises: retrieving actual
deformation and bending moment measurements for spaced-apart intervals on a bottomhole
assembly (BHA) as a function of BHA position; obtaining an estimated borehole trajectory;
deriving predicted deformation and bending moment measurements based on the estimated
borehole trajectory; determining an error between the predicted measurements and the actual
measurements; updating the estimated borehole trajectory to reduce the error; repeating said
deriving, determining, and updating to refine the estimated borehole trajectory; converting the
estimated borehole trajectory into a borehole curvature log for display or storage on a
nontransient information storage medium.
Each of the foregoing embodiment may further include any of the following additional
elements alone or in any suitable combination: 1. The method includes displaying the borehole
curvature log. 2 . The method includes storing the borehole curvature log on a nontransient
information storage medium. 3 . The BHA further includes navigation sensors, and wherein
said obtaining includes processing measurements from the navigation sensors. 4 . Said deriving
includes performing a three-dimensional finite element analysis to determine curvature of the
BHA based on the estimated borehole trajectory. 5 . Said deriving includes employing a stiffstring
model to determine curvature of the BHA based on the estimated borehole trajectory. 6 .
Said deriving includes determining side forces on the drill string, including drag and torque. 7 .
Said deriving accounts for tool joint dimensions and spacing. 8 . The borehole curvature log
specifies, as a function of position along the borehole, a rate at which the borehole trajectory
changes in degrees per unit length. 9 . The borehole curvature log specifies, as a function of
position along the borehole, a radius of curvature.
Numerous other modifications, equivalents, and alternatives, will become apparent to
those skilled in the art once the above disclosure is fully appreciated. It is intended that the
following claims be interpreted to embrace all such modifications, equivalents, and alternatives
where applicable.

CLAIMS
WHAT IS CLAIMED IS:
1. A borehole curvature logging system that comprises:
a drill string having a bottomhole assembly (BHA) with sensors providing actual deformation
and bending moment measurements as a function of BHA position at spaced-apart intervals
on the BHA;
a processing system that retrieves said actual measurements and responsively generates a log
of borehole curvature by:
providing an estimated borehole trajectory;
deriving predicted deformation and bending moment measurements based on the
estimated borehole trajectory;
determining an error between the predicted measurements and the actual measurements;
updating the estimated borehole trajectory to reduce the error;
repeating said deriving, determining, and updating to refine the estimated borehole
trajectory; and
converting the estimated borehole trajectory into a borehole curvature log; and
a user interface that displays the borehole curvature log.
2 . The system of claim 1, wherein the BHA further includes navigation sensors, and wherein
said providing is based in part on measurements from the navigation sensors.
3 . The system of claim 1, wherein as part of said deriving, the processing system employs a
three-dimensional finite element analysis to determine curvature of the BHA based on the
estimated borehole trajectory.
4 . The system of claim 1, wherein as part of said deriving, the processing system employs a
stiff-string model to determine curvature of the BHA based on the estimated borehole
trajectory.
5 . The system of claim 1, wherein as part of said deriving, the processing system determines
side forces on the drill string, including drag and torque.
6 . The system of claim 1, wherein as part of said deriving, the processing system accounts for
tool joint dimensions and spacing.
7 . The system of claim 1, wherein the borehole curvature log specifies, as a function of position
along the borehole, a rate at which the borehole trajectory changes in degrees per unit length.
8 . The system of claim 1, wherein the borehole curvature log specifies, as a function of position
along the borehole, a radius of curvature.
9 . A borehole curvature logging method that comprises:
retrieving actual deformation and bending moment measurements for spaced-apart intervals
on a bottomhole assembly (BHA) as a function of BHA position;
obtaining an estimated borehole trajectory;
deriving predicted deformation and bending moment measurements based on the estimated
borehole trajectory;
determining an error between the predicted measurements and the actual measurements;
updating the estimated borehole trajectory to reduce the error;
repeating said deriving, determining, and updating to refine the estimated borehole traj ectory;
converting the estimated borehole trajectory into a borehole curvature log for display or
storage on a nontransient information storage medium.
10. The method of claim 9, further comprising displaying the borehole curvature log.
11. The method of claim 9, further comprising storing the borehole curvature log on a
nontransient information storage medium.
12. The method of claim 9, wherein the BHA further includes navigation sensors, and wherein
said obtaining includes processing measurements from the navigation sensors.
13. The method of claim 9, wherein said deriving includes performing a three-dimensional
finite element analysis to determine curvature of the BHA based on the estimated borehole
trajectory.
14. The method of claim 9, wherein said deriving employs a stiff-string model to determine
curvature of the BHA based on the estimated borehole trajectory.
15. The method of claim 9, wherein said deriving includes determining side forces on the drill
string, including drag and torque.
16. The method of claim 9, wherein said deriving accounts for tool joint dimensions and
spacing.
17. The method of claim 9, wherein the borehole curvature log specifies, as a function of
position along the borehole, a rate at which the borehole trajectory changes in degrees per unit
length.
18. The method of claim 9, wherein the borehole curvature log specifies, as a function of
position along the borehole, a radius of curvature.