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Mud Motor PDM Dynamics: An Analytical Model

Conference Paper · September 2021

DOI: 10.2118/206064-MS

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IADC/SPE-208789-MS

Mud Motor PDM Dynamics: A Control Model for Automation

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Robello Samuel, Halliburton; Fedor Baldenko, Gubkin University; Dmitry Baldenko, VNIIBT

Copyright 2022, IADC/SPE International Drilling Conference and Exhibition DOI 10.2118/208789-MS

This paper was prepared for presentation at the IADC/SPE International Drilling Conference and Exhibition held in Galveston, Texas, USA, 8–10 March 2022.

This paper was selected for presentation by an IADC/SPE program committee following review of information contained in an abstract submitted by the author(s).
Contents of the paper have not been reviewed by the International Association of Drilling Contractors or the Society of Petroleum Engineers and are subject to correction
by the author(s). The material does not necessarily reflect any position of the International Association of Drilling Contractors or the Society of Petroleum Engineers,
its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the International Association of Drilling
Contractors or the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations
may not be copied. The abstract must contain conspicuous acknowledgment of IADC/SPE copyright.

Abstract
Automating drilling operations using modern technology with intelligent control systems is prioritized with
respect to drilling process development. The process becomes more complex when a mud motor is used
because of the dynamic properties of the control objects, which are interconnected from the surface feed
mechanism to the bit. Development of a model that better predicts the dynamic condition when a mud motor
is used is discussed. The dynamic characteristics of a positive displacement motor (PDM) are considered
as a single system by coupling the hydromechanical processes. This includes, transient pressure, drillstring
dynamics, kinematics of the PDM rotor, the mud pump and dampener, dynamic characteristics of the bit, and
internal and external system disturbances. The mathematical model includes the hydraulic and mechanical
subsystems as well as the relationships of these subsystems. The subsystems include the equations of the
processes in their constituent links (i.e., drillstring, hydraulic line, mud pump, downhole motor, bit, and bit
feed mechanism). The nonlinear system of differential equations are solved using numerical methods with
appropriate boundary conditions in a two-way loop for regulating the load on the bit and flowrate.
The study shows that the transient behavior of the mud motor must be accounted for when automating the
drilling process. It has been observed that an instantaneous change in the load results in the transition of the
hydraulic motor to a new steady-state mode gradually (for tens of seconds) and is influenced by the transient
pressures in the string. The transient process (torsional vibrations) in the PDM occurs until the flow rate
stabilizes at the top of motor. A similar hydromechanical effect, if the PDM does not have a sufficient torque
reserve, can lead to deceleration of the PDM and might require correction of the weight on bit (WOB) and
flow rate. The study also showed that the effect of pulsation dampener resulting in an uneven flow leads to
uneven rotation of the mud motor shaft, even if a constant load on the bit is maintained. This study revealed
that evaluation of the influence of the deviation of one of the operating parameters on the behavior of the
dynamic system is important (Tokhonov, 2019). The choice of optimal control algorithms depends on the
accuracy of determining the transfer functions, in particular, with respect to the change in the WOB. Further
transfer functions derived can be used to control the surface parameters for drilling.
2 IADC/SPE-208789-MS

Introduction
Amongst all types of downhole motors, PDMs have become more widespread. The history of these hydraulic
machines begins in 1930 when the French engineer R. Moineau patented the original scheme of a volumetric
rotary hydraulic machine. In the 1960s, in the US and Russia, as a result of research and development
work on the basis of this scheme, PDMs were created for drilling wells with various lobe configurations
and designs. Currently, PDMs are manufactured by several engineering and service companies in the US,

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Canada, Europe, Russia, and China (diametral dimensions from 43 to 288 mm). The PDM successfully
competes with the rotary drilling method and is a leader amongst other types of downhole motors, surpassing
them in such important indicators as specific torque and power, speed stability under variable loads, as well
as design simplicity and low detail. At the same time, the peculiarity of the PDM operating principle, because
of the eccentricity of the rotor movement inside the stator, leads to the fact that such a motor becomes an
additional source of vibration excitation in the general dynamic system.
In a fast-drilling environment, such as shale drilling, refining advanced technologies for preventing
downhole tool failures is paramount. Challenges are still greatly associated with complex bottomhole
assemblies and the vibration of the drillstring when used with a downhole mud motor (Samuel et al.
2021). When automating drilling, including sliding and rotary modes with a downhole motor, additional
methodology should be included so the system is in stable equilibrium. Unbalanced force attributed to the
eccentric mass of the rotor results in tangential and normal forces. These projections are considered and
integrated into the full drillstring forced frequency model as force and displacement at the motor location.
The model also considers the effect of the string speed. The unbalanced force is more pronounced at the
lower pair or lobes configuration compared to the higher pair lobe configuration because of the larger
eccentricity (Samuel et al. 2018a, 2018b and 2020). The unbalance is modeled in terms of an equivalent mass
of the rotor with the eccentricity of the rotor. Additionally, the analysis provides an estimation of relative
bending stress, shear force, and lateral displacement for the assembly used. Based on the study, severe
vibrations causing potentially damaging operating conditions can be avoided once the motor operating
envelope can be estimated (Samuel 20018a and 2018b).

Digital Solution
Well construction in the digital format provides planning, designing, and well construction using
collaborative well engineering and drilling automation. It provides the option to minimize planning cycle
times, reduce drilling costs, and accelerate critical decision-making to help mitigate risk, maximize reservoir
contact, and achieve repeatable drilling performance, as shown in Fig. 1. The drilling plan and execution
changes when the drillstring has a mud motor or rotary steerable system. The integrated approach makes it
easy to monitor and control the operation in real time, manually or automatically, providing the following:

• Standardization of commands to the PDM system and activities through automation.

• Time-efficient lean process engine to plan, design, and execute wells.

• A single point of access to all data, kept consistent and available securely to all disciplines.

• Rapid multidisciplinary and multivendor collaboration on an open platform so that any mud motor
from the fleet can be connected.
• Intelligent, engineer-driven rule-based decision making so that mud motor life is increased.

• Rapid extraction of design reports for a final well design.

IADC/SPE-208789-MS 3

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Figure 1—Well construction digital framework for automation.

Furthermore, the framework enables monitoring execution of the slide sheet, a "live" plan and leverage
automated workflow that facilitates re-engineering of the well and rig activity program in minutes and
enhances the asset management from the past bit-motor-bottomhole assembly (BHA) response.
By connecting the entire value chain in real time and leveraging automated workflows, operators can
optimize their well operations to help reduce costs, enhance production, and increase the overall ROI.

Model Formulation
The methodology is necessary to describe the mechanical system so that controls and setpoints are developed
properly. The mechanical system and the interconnected components referenced in Fig. 1 are shown in Fig.
2. The mechanical system that must be included in the modeling are:

• Bit-motor-BHA component (bit – output shaft – mud motor).

• Long drillstring.

• Lumped mass that includes swivel, hook, traveling block, topdrive motor.

• Drawworks and control parameters (manual or auto driller).

4 IADC/SPE-208789-MS

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Figure 2—Mechanical subsystem.

Different components are:

• Drawworks.

• Tackle system.

• Hook, swivel.

• Drillstring.

• Downhole motor.

• Bit.

Surface-Bit Regulator
The equation of motion of the drawworks for moving the drillstring is given as

(1)

where
Jd is reduced moment of inertia of the rotating parts of the drawworks during drilling.
Td = torque at the drawworks drum, which is given a

(2)
here
Fd is tension force of the running string of the rope on the drum of the drawworks.
Ddr is the average diameter of drill line on the drawworks drum.
IADC/SPE-208789-MS 5

Tbr,d = braking torque at the drawworks drum.

The instantaneous angular velocity of the drum is given as:

(3)

vd is the linear speed of the running string of the rope.

The generalized equation of motion of the upper end of the drillstring can be given as:

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(4)

where
v(0); F(0) is the speed and the axial force at the upper section.
m∑ is the weight of swivel, traveling block, topdrive motor.
is the sum of the efforts in the working strings of the rope interacting with the pulleys of the
traveling block; n is the number of traveling block pulleys.
Further, the generalized equation of motion of the upper end of the drillstring can be modified as

(5)

where
m* is the reduced mass of the translationally moving and rotating masses block and tackle system and is
given as:

(6)

its, jts - kinematic and power transmission ratio of the tackle mechanism, variable in time.

Drilling Mode Control System

Automated control systems with mud motors involve a combination of advanced dynamic mathematical
modeling, microprocessor technology, and a control system so that intelligent decision-making can occur
in real time. The dynamic modeling should include an interconnected system of hydraulic and mechanical
systems and a mud pump system, including a surface line, drillstring, downhole motor, and bit. In theoretical
terms, the complexity of system design and development of algorithms for controlling the automated drilling
takes into consideration the bottomhole disturbance and the control system because of the lag time between
these two end points. Therefore, a methodology is necessary for forward predictive estimation in the control
algorithms. Additionally, because the control object is a system with an unknown time-varying dynamic
characteristic, it is necessary to use adaptation methods during control.
In general, the axial load on the bit W depends on the longitudinal force in the lower section (x = l) of
the drillstring:
(7)
where
Gdm is the downhole unit gravity.
α - angle of deviation of DHM from the vertical (zenith angle at the bottom of the well).
Z is the axial hydraulic force due differential pressure in the motor and bit.
When controlling the downhole mud motor with the automated control system for drilling the regulation
parameter that can be used are hook load or topdrive axial speed, standpipe pressure, mud pump speed, and
mud motor differential pressure (using telemetry system). In the known methods of controlling the load on
6 IADC/SPE-208789-MS

the bit, based on maintaining a constant force or speed of the upper section of the drillstring (F (0) = const;
v(0) = const), the dynamic state of the system (as a deformable rod with moving ends) is determined by the
combined action of tensile stress (compression) and torsion, distributed along the length of the string. Force
factors (longitudinal load and torque) in each column section x depend on the corresponding deformations
(∂u/∂x;∂θ / ∂x) in the sections given by

(8)

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When calculating dynamic processes and modeling the control system, it is essential to consider the
combined action of compressive (tensile) and torsion stresses distributed along the pipe length. In the
idealized static mode when drilling, the velocities of the upper and lower sections of the string are the same
(v(0) = v(l)) and correspond to the ROP proportional to the drillability ß and the axial load on the bit (ROP
= vh = βW). In this case, the longitudinal forces in the upper and lower sections of the drillstring, which
determine the tension force at the hook given as Wh = F(0) and the load on the bit W, respectively, differ by
the value of the longitudinal component of the buoyed weight of the string Gcol x.
Considering the friction force Ff, along the wellbore wall
(9)
Estimating the friction force Ffr is difficult during the drilling process because of the uncertainty of the
wellbore position in deviated and horizontal wells, which makes it difficult to apply the required axial load
to the bottom and maintain the specified mode of mud motor operation.
In actual drilling conditions, because of the elasticity of the pipe when the pipe is moving, instantaneous
velocities at the boundary condition (i.e., upper (ds/dt) and lower (dh/dt) ends) are not the same, and it varies
from time t (Fig. 3.a) to t + dt (Fig.3.b) with the instantaneous movement of the section dz, relative to the
fixed coordinate system. This will result in a different displacement at the hook and at the bit (Fig. 3). Eq. 9
does not reflect the actual dynamic processes for the long string because the force factors in Eq. 8 in various
sections become dependent on wave processes (∂u/∂x; ∂θ/∂x = var).
IADC/SPE-208789-MS 7

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Figure 3—Dynamic movement of the drillstring.

If one assumes that the change in the linear deformation of all sections of the column is the same (spring
model), then the dependence on the instantaneous velocities of the end sections of the column is reduced
to an approximate form as given next:
(10)
Evaluation of the influence of the deviation of one of the previous operating parameters on the behavior
of the dynamic system and the choice of optimal control algorithms depends on the accuracy of determining
the transfer functions, particularly with respect to change in the WOB ((ΔW):
The transfer functions are classified
• by the specific moment of formation:

(11)

• by hook load:

(12)

• by drilling mud consumption:

(13)
• by the frictional force of the drillstring:
(14)
Where
ε - dimensionless parameter that determines the static characteristics of the system "drillstring - motor
- bit - formation" (load transfer coefficient):
8 IADC/SPE-208789-MS

(15)
ψ is a dimensional parameter expressing the change in the longitudinal force on the motor body when
the flow rate changes (flow rate transfer coefficient):

(16)

A is the flow area of the drillpipes.

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From the presented expressions of the transfer functions, it follows that the main factor determining the
response of the considered dynamic system to the deviation of the parameters of its state from the nominal
(initially established) values is the dimensionless complex ε = Tsp AKP composed of formation dimensional
parameters (Tsp), drillpipe (A), and motor characteristic pressure drop slope (kP). This complex is a criterion
for the similarity of a hydromechanical system, expressing its load characteristics in the case of using a
hydraulic motor with a variable pressure drop. The physical meaning of this similarity criterion (ε <1) is
the ratio of the increment in the longitudinal force on the motor body to the increment in the axial load on
the bit during the drilling of these formations.
For example, when drilling with mud motor with a bit diameter of 195 mm (7.67 in.) (kP = 0.7 kPalN•m,
0.138 psi/(lbf × ft)) using 127 × 9 (5 × 0.35 in.) drillpipes and roller cone bits in medium-hard formations
(Tsp = 10 N•mlkN) ε = 0.065. This means that, when the hook load is reduced by 10 kN (because of improper
actions by the driller or a failure in the control system), the WOB will increase by 10.7 kN, and, when
moving into the soft formation layer (Tsp = 15 N•mlkN), the WOB will increase by 3.6% (because of the
increase in hydraulic force from the pressure drop across the PDM).
Fig. 4 presents graphical dependences of transfer functions, which can be used as nomograms to identify
operating parameters or when designing automated control systems for the well drilling process. Oscillatory
processes in the pressure hydraulic line, which affect the uniformity of rotation of the downhole motor
and the quality of operation of telemetric systems for monitoring downhole parameters with a hydraulic
communication, depend on a combination of two similarity criteria: frequency μ and acoustic wave travel.

Figure 4—Transfer functions of the dynamic system "drillstring - PDM - bit – formation."

In the case of equipping drilling rigs with a variable drive, it is possible to tune the "pump-pressure line-
downhole motor" system to stabilize the hydrodynamic regime (to help ensure a minimum uneven pressure
at the pump outlet and fluid flow rate at the mud motor inlet) to reduce energy consumption for hole cleaning,
increasing the durability of downhole and surface equipment, increasing the ROP and penetration to the bit,
and reducing the level of interference in the well during signal transmission when using telemetry systems
with a hydraulic communication channel.
IADC/SPE-208789-MS 9

Digital Control with PDM

The transfer functions must be connected to the drillahead predictive model, which can be used in the
planning stages as well as during operational stages. This operational parameters can be calculated so that
analysis can be used to predict the directional behavior of a BHA during the planning stages. With this

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option, it is possible to study the effects of various components, including bent assemblies, collar sizes,
stabilizer placement, hole enlargement, and component wear when the mud motor is used. During well
operations, it can be used to adjust surface operating parameters to optimize performance. The drill-ahead
analysis first performs the static analysis so that the surface and downhole parameters are calibrated. The
program then drills ahead in defined foot increments to predict the BHA behavior over the specified drill-
ahead interval. This analysis option also can be helpful for determining the optimal WOB/ROP combination.
Further, the full 3D BHA model combined with the PDM allows:

• Predicting the directional behavior (including build, walk, and drop) of a BHA as it drills ahead
through a specified interval.
• Predicting the transient effect when new assembly is run in hole.

• Adjusting operating parameters to affect BHA performance.

• Studying effects of bent assemblies, collar size, stabilizer placement, eccentric stabilizers, stabilizer
wear, hole enlargement, and operating parameters for optimal performance.
• Selecting proper bent subs to achieve desired build or drop rates.

• Estimating the additional torque drawn from a motor attributed to lateral forces at bit.

• Distinguishing between steady state and transient behavior.

• Determining the downhole mechanism controlling the BHA.

• Determining the orientation of a BHA (0 - 180° left or right of high side) for achieving optimum
performance in a well deflection scenario.
• Comparing a rotary versus steerable assembly performance for a given well trajectory analysis.

• Optimizing the design of a steerable system through modeling of number of bends and eccentric
contact points in the BHA.
The calculation with the combined PDM performance and BHA tendency provide options to calculate the
side forces at various components, including at the bit, build rate, and walk rate. Other calculations at the bit
include wellbore angle, wellbore inclination relative to the vertical plane, inclination of the string or bit face
relative to the vertical plane, bit tilt, magnitude of the force in the inclination plane, acting perpendicular
to the bit, the hole direction angle relative to the direction plane, (the direction plane is 90° to the vertical
plane, and 90° to the string axis), string direction angle relative to the direction plane, (direction plane is 90°
to the vertical plane, and 90° to the string axis), angle of the bit relative to wellbore angle in the direction
plane, the magnitude of the force in the direction plane, and acting perpendicular to the bit.
10 IADC/SPE-208789-MS

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Figure 5—Change in ROP and baseline current Iset.

A feature of the drilling control system using BFR based on an electric powder brake is its asymmetric
response to the surge (+ΔU) and release (-ΔU) of the brake voltage, which is explained by the continued
drilling out of the formation and a gradual decrease in the load on the bit when the winch drum is braked,
because in-passive BFR lifting of the tool, or reverse rotation of the lifting shaft, is not possible.
When controlling the drilling mode of a hydraulic motor with a variable pressure drop, it is necessary to
consider the hydromechanical effects in a positive feedback system (drillstring - PDM - bit), in which an
increase in torque (e.g., when the bit enters an interlayer of more torque-intensive formations) leads to an
increase in the pressure drop in the PDM, which, in turn, is accompanied by lengthening of the drillstring
and, as a consequence, leads to a corresponding increase in the axial load on the bit and an even greater
increase in torque. A similar hydromechanical effect, if the PDM does not have a sufficient torque reserve,
can lead to deceleration of the PDM and require correction of the target WOB and fluid flow rate.

Conclusions
• Based on the study, inclusion of a mud motor plays an important role, not only during the planning
stage, but also during drilling. The static and dynamic characteristics of the motor heavily influence
the drilling process operating parameters.
• This study provides insight into the motor dynamics of various downhole assembly configurations.
Vibrational intensity provides an option to estimate the critical zones to be avoided.
• Also provided is a clear presentation of the dynamic roadmap so that the proper motor operational
parameters can be used.
• Hydrodynamical effects coupled with trajectory control are necessary for proper well placement.

• Transfer functions of the dynamic system "drillstring – PDM – bit – formation" proposed can be
used to control the surface parameters of a complex drilling system.
• The choice of optimal control algorithms depends on the accuracy of determining the transfer
functions, particularly with respect to change in the WOB.

References
Samuel, R., Azar, J.J., Aideyan, P. 2018a. Applied Drilling Engineering Optimization. SigmaQuadrant Publication. http://
www.sigmaquadrant.com.
IADC/SPE-208789-MS 11

Samuel, R., Baldenko, D., Baldenko, F. 2018b. Positive Displacement Motor: Theory and Applications. SigmaQuadrant
Publication. http://www.sigmaquadrant.com.
Samuel, R., Baldenko, D., Baldenko, F. 2020. Mud Motor (PDM) and Well Engineering: Comprehensive Guide.
SigmaQuadrant Publication. http://www.sigmaquadrant.com.
Samuel, R., Baldenko, F., Baldenko, D. 2021. Mud Motor PDM Dynamics: An Analytical Model. Presented
at the SPE Annual Technical Conference and Exhibition, Dubai, UAE, September. SPE-206064-MS. https://
doi.org/10.2118/206064-MS.
Tikhonov, V.S., Baldenko, F.D., Bukashkina, O.S. et al 2019. Effect of Hydrodynamics on Axial and Torsional Oscillations

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of a Drillstring with using a Positive Displacement Motor. International Journal of petroleum Science and Engineering.
183:106423. DOI:10.1016/j.petrol.2019.106423.

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