# Autonomous Decision-Making While Drilling

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## Abstract

**:**

## 1. Introduction

## 2. Problem Scope

- Land rigs, fixed platforms, or floaters;
- Any depth ranges and well shapes;
- Simple or tapered drill-strings and wellbore architectures;
- Under-reaming or hole opening operations;
- Water-based, oil-based, or synthetic-based drilling fluids, but not foams as this would lead to a dual gradient drilling operation;
- Complex hydraulic networks with multiple paths such as leakage to the annulus from an under-reamer, hole opener or positive displacement motor, or the use of booster pumping in a riser;
- Low and high bandwidth communication methods, including possibly distributed sensors along the drill-string.

- The time to execute the series of actions;
- The time needed to mitigate any drilling incidents if they occur.

## 3. State Estimation and Uncertainty

- Mass conservation for the drilling fluid [17]: $\frac{\partial {\rho}_{m}}{\partial t}+\nabla .({\rho}_{m}{\overrightarrow{u}}_{f})=0$, where ${\rho}_{m}$ is the drilling fluid density, $t$ is time, ${\overrightarrow{u}}_{f}$ is a fluid velocity vector;
- Momentum conservation for viscous flow (Navier-Stokes) [17]: ${\rho}_{m}\left(\frac{\partial {\overrightarrow{u}}_{f}}{\partial t}+\left({\overrightarrow{u}}_{f}.\nabla \right){\overrightarrow{u}}_{f}\right)=-\nabla p+\nabla .\stackrel{=}{\tau}+{\rho}_{m}\overrightarrow{g}+{\rho}_{m}{\overrightarrow{f}}_{b}$, where $p$ is pressure, $\stackrel{=}{\tau}$ is the stress tensor, $\overrightarrow{g}$ is the gravitational acceleration and ${\overrightarrow{f}}_{b}$ represents external body force;
- Force balance on particles/bubbles transported by the fluid (Newton) [18,19]: ${\rho}_{f}{V}_{p}\frac{d{\overrightarrow{u}}_{p}}{dt}={\overrightarrow{F}}_{p}$, where ${\rho}_{f}$ is the density of the background fluid, ${V}_{p}$ is the particle volume, ${\overrightarrow{u}}_{p}$ is the particle velocity vector, ${\overrightarrow{F}}_{p}$ is the external force vector applying on the particle;
- Torque balance on particles/bubbles transported by the fluid (Newton): $\frac{{\partial}^{2}\left({\rho}_{p}{I}_{p}{\overrightarrow{\omega}}_{p}\right)}{\partial {t}^{2}}={\overrightarrow{M}}_{p}$, where ${\rho}_{p}$ is the particle density, ${\overrightarrow{\omega}}_{p}$ is the angular velocity of the particle, ${I}_{p}$ is the second moment of area and ${\overrightarrow{M}}_{p}$ is an external torque applying on the particle;
- Energy conservation for heat transfer (Fourier) [20]: $\frac{\partial}{\partial t}\left({\rho}_{m}H\right)-\nabla \left({Q}_{f}+{Q}_{c}\right)-{q}_{s}=0$, where $H$ is the enthalpy per mass unit, ${Q}_{f}$ is the forced convective term, ${Q}_{c}$ is the conductive and natural-convective term, ${q}_{s}$ is the heat generated by mechanical and hydraulic frictions. The enthalpy can be expressed as a function of temperature $T$ and pressure by the following expression: $dH={C}_{p}dT+V\left(1-\alpha T\right)$, where ${C}_{p}$ is the specific heat capacity, $V$ is the volume, and $\alpha $ is the volumetric coefficient of thermal expansion;
- Force balance for elastic deformation of the drill-string (Newton) [21]: $\frac{\partial \overrightarrow{T}}{\partial s}+{\overrightarrow{f}}_{s}={\rho}_{s}A\frac{{\partial}^{2}{\overrightarrow{u}}_{s}}{\partial {t}^{2}}$, where $\overrightarrow{T}$ is the internal tension vector in the solid, $s$ is a curvilinear abscissa, ${\overrightarrow{f}}_{s}$ is an external force per unit length, ${\rho}_{s}$ is the density of solid constituting the string, $A$ is an area and ${\overrightarrow{u}}_{s}$ is the velocity of control element of a portion of string;
- Torque balance for elastic deformation of the drill-string (Newton) [21]: $\frac{\partial \overrightarrow{M}}{\partial s}+\overrightarrow{t}\times \overrightarrow{T}+\overrightarrow{c}=\frac{{\partial}^{2}\left({\rho}_{s}{I}_{s}{\overrightarrow{\omega}}_{s}\right)}{\partial {t}^{2}}$, where $\overrightarrow{M}$ is the internal torque in the solid, $\overrightarrow{t}$ is the tangential vector of the Frenet–Serret coordinate system, $\overrightarrow{c}$ is an external torque, ${I}_{s}$ is the second moment of area, and ${\overrightarrow{\omega}}_{s}$ is the angular velocity of a control element of a portion of a string;
- Energy conservation for linear deformation of the drill-string (Euler-Bernoulli) [22]: $\frac{{\partial}^{2}}{\partial {s}^{2}}\left(EI\frac{{\partial}^{2}w}{\partial {s}^{2}}\right)+\mu \frac{{\partial}^{2}w}{\partial {t}^{2}}=q$, where $w$ is a deflection in a perpendicular direction to $\overrightarrow{t}$, $E$ is the elastic modulus, $\mu $ is a mass per unit length, and $q$ is the potential energy of external loads.

## 4. Protection Layers

- Protection of the commands sent to the drilling machines, also referred to as safe operating envelopes,
- Protection of the drilling process, i.e., automatic fault detection, mitigation, and recovery (FDMR);
- Protection of the process during transition from autonomous to manual control by automatic management of safe operational modes.

#### 4.1. Safe Operating Envelopes

- Axial velocity;
- Flowrate;
- Rate of penetration (ROP).

#### 4.1.1. Axial Velocity

- If there is no circulation, the limits are computed as a function of the gel duration.
- If there is circulation, the limits are computed as a function of the drill-string rotational speed and flowrate.

#### 4.1.2. Flowrate

- Drill-string axial velocity and heave levels;
- Drill-string rotational velocity;
- Bit and bottom hole depths;
- Temperatures;
- Cuttings load.

#### 4.1.3. ROP

- Formation unconfined compressive strength (UCS);
- Formation angle of internal friction;
- A parameter related to the cutting forces orientation (proxy for bit aggressiveness);
- Threshold for transition between phase 1 and 2 of the bit response;
- Threshold for transition between phase 2 and 3 of the bit response (founder point);
- Two parameters related to the variation of frictional force with WOB in phase 1 and 3, respectively. The latter influences the stagnation or decrease in ROP as WOB is added beyond the founder point.

- Onset of drill-string sinusoidal buckling;
- Poor cuttings transport, leading to formation of cuttings beds;
- Excessive cuttings concentration in suspension, leading to pack-offs;
- Exceeding of geo-pressure margins, defined by the pore, collapse, and fracture pressure, and minimum horizontal stress;
- Excessive mud-pump pressure.

#### 4.2. Fault Detection, Mitigation and Recovery

#### 4.2.1. Overpull/Set-Down Weight

#### 4.2.2. Over-Torque

#### 4.2.3. Overpressure

#### 4.2.4. Drilling Dysfunctions

#### 4.3. Safe Mode Management

## 5. Drilling Procedures

- A process model: it is a hierarchical decomposition of the overall process into process stages, which are themselves subdivided into process operations, which make use of the process actions.
- A physical model: it is also a hierarchical categorization of the overall enterprise, into sites which consist of areas, themselves containing process cells, in which there may be units, themselves composed of equipment modules, which finally may be made of control modules.
- A procedural control model: it describes how the batch process should be carried out. It is also hierarchically organized with the first level subcategory being a unit procedure, the second hierarchical level being an operation, which itself is made of phases.

- Should we use two out of three mud pumps so that the third one can be used as a booster pump to clean the marine risers?
- Should the mud pumps be started with different pump rates in order to minimize the risk of accentuated stroke noise that can perturb the decoding of mud pulses from downhole telemetry?

- Contextual level, i.e., drilling a 12 ¼-in section,
- Conceptual level, i.e., drilling one stand,
- Logical level, i.e., running a friction test,
- Physical and detailed level, i.e., unwinding the drill-string to reach zero torque after stopping the top-drive.

#### 5.1. Logical Level

- Take off slips: when the drill-string is in slips, it is lifted to transfer the weight from the slips to the top-drive.
- Top-drive startup: the top-drive may be ramped up in one or several stages. The accelerations and stages should be chosen carefully to limit the risk for intense drill-string vibrations, at least in deviated wells.
- Mud pump startup: first the air gap at the top of the drill-string needs to be filled without using too much time. Then circulation must be established. This means breaking the gel and reaching a steady flow into the whole hydraulic circuit. Breaking the gel may be assisted by rotating the drill-string. Then the mud pump rate is increased toward its nominal value. This can be done in one or several steps depending on the operational risks for pack-offs or differential sticking. It may be necessary to stop the top-drive, if it was running, and a survey shall be taken. If the top-drive has been stopped, it may be necessary to lift up and down the drill-string several times in order to remove some of the trapped torque, at least for deviated wells. When mud pulse telemetry has been established, the top-drive may be started.
- Tag bottom: on a floater, the heave compensator may need to be started. Furthermore, the axial tagging velocity shall be chosen to give a clear signal that the bit is on bottom and yet not be the source of large stick-slips because of a large step-change in WOB when touching the bottom hole.
- Drill: the drilling parameters, i.e., WOB, top-drive speed, and flowrate, should be continuously adapted to the current drilling conditions both to optimize the penetration rate but also control risks of drill-string vibrations and poor cuttings transport.
- Reciprocate: to improve the hole conditions it may be necessary to ream-up and down for a certain distance. The choice of top-drive speeds, axial velocities, and flowrates should be adapted to the current downhole conditions and potential risks for pack-offs and surging.
- Perform a friction test: if information about the downhole mechanical friction is necessary, a pick-up and a slack-off procedure may be performed. The pick-up distance and the axial velocities should be chosen as a function of the length of the drill-string, expected friction, and possible consequences on swab and surge pressures. The flowrate may also have to be adjusted for performing the friction test.
- Move to stick-up height: on a floater it may be necessary to stop the heave compensator if it was turned on. Then the drill-string is lowered such that the tool-joint is at the correct level for the iron roughneck.
- Stop top-drive: the top-drive rotation speed is brought down to zero with a controlled deceleration. With deviated wells or if heave variations are low, it may be necessary to apply a zero-torque procedure consisting of unwinding the drill-string to remove the torque that is still trapped in the drill-string because of the mechanical friction.
- Stop mud pumps: the mud pumps are ramped down with a controlled deceleration procedure in order to avoid large downhole pressure variations due to the drilling fluid inertia effects.
- Set in slips: the slips are closed, and the drill-string is lowered to transfer the weight from the top-drive to the slips.
- Perform booster pumping: during any procedural operations where the mud pumps are active, it may be necessary to pump from the bottom of the riser in order to assist lifting cuttings that are trapped in the riser.

#### 5.2. Physical and Detailed Levels

## 6. Decision Making and Risk Mitigation

- Perform a friction test: this provides possibly useful insight about the quality of the hole being drilled but does not contribute directly to the hole creation process, and can be a source of risk as lifting the pipes up and down without rotation can lead to overpulls, set-down weights that can lead to stuck pipe situations.
- Perform a reciprocation: this improves the downhole conditions by displacing the cuttings upward. Additionally, torque analysis while reciprocating is a good source of information for the estimation of the wellbore state, and downhole measurements are still available during this operation since high circulation is established. This action does not contribute directly to the drilling process and has therefore a cost.
- Drill, with a possible speed reduction: reducing the speed may limit the deterioration of the downhole conditions or even improve them. This is of course at the cost of drilling performance. Also, since the bit is on-bottom, surface measurement analysis (hook-load and surface torque) is much more difficult, and one mainly relies on downhole pressure to estimate the wellbore quality.

#### 6.1. Actions

- ${r}_{d}={f}_{d}=0$ if the block is higher than 15 m,
- ${r}_{c}={f}_{c}=c=0$ if the block is higher than 5 m,
- ${r}_{d}={f}_{d}=0$ and $c=1$ if the block is lower than 5 m.

#### 6.2. States

#### 6.3. Transition Functions

#### 6.4. Reward/Penalty Function

#### 6.5. Solution

## 7. Results and Discussions

#### 7.1. Examples

#### 7.2. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A$ | area (m^{2}) |

$A$ | set of possible actions |

$a$ | an action |

$a$ | bias parameter of the friction function |

$b$ | slope parameter of the friction function |

$C$ | a cost function |

${C}_{p}$ | specific heat capacity (J/(kg·K)) |

$c$ | a cost |

$c$ | indicator of the occurrence of a connection |

$\overrightarrow{c}$ | external torque (N·m) |

$d$ | pace reduction applied to the ROP (s/m) |

$E$ | elastic modulus (Pa) |

${\overrightarrow{F}}_{p}$ | external force vector applying on the particle (N) |

${\overrightarrow{F}}_{\mu}$ | total kinetic friction force vector (N) |

${F}_{{\mu}_{k}}$ | limit of the kinetic friction force at high velocity (N) |

${F}_{{\mu}_{s}}$ | upper limit of the static friction force (N) |

${\overrightarrow{f}}_{b}$ | external body force (N/kg) |

${f}_{c}$ | indicator for friction test procedure at connection time |

${f}_{d}$ | indicator for friction test procedure while drilling |

${\overrightarrow{f}}_{s}$ | external force per unit length (N/m) |

$\overrightarrow{g}$ | gravitational acceleration (m/s^{2}) |

$H$ | enthalpy per mass unit (J/kg) |

${I}_{p}$ | second moment of area for particle (m^{4}) |

${I}_{s}$ | second moment of area for string (m^{4}) |

$\overrightarrow{M}$ | internal torque in the solid (N·m) |

${\overrightarrow{M}}_{p}$ | external torque applying on the particle (N·m) |

$\widehat{n}$ | normal unit vector at the contact (dimensionless) |

$p$ | pressure (Pa) |

${Q}_{c}$ | conductive and natural-convective term (W) |

${Q}_{f}$ | forced convective term (W) |

$q$ | potential energy of external loads (J) |

${q}_{s}$ | heat generated by mechanical and hydraulic frictions (W/m) |

$\overrightarrow{R}$ | reaction force between the surfaces in contact (N) |

${r}_{c}$ | indicator for reciprocation procedure at connection time |

${r}_{d}$ | indicator for reciprocation procedure while drilling |

$S$ | set of system states |

${S}^{\prime}$ | a subset of $S$ |

$s$ | curvilinear abscissa (m) |

$s$ | a state |

${s}^{\prime}$ | a new state |

$T$ | density function for a distribution of state |

$T$ | temperature (K) |

$\overrightarrow{T}$ | internal tension vector in the solid (N) |

$\overrightarrow{t}$ | tangential vector of the Frenet-Serret coordinate system (dimensionless) |

$t$ | time (s) |

${U}^{\pi}$ | expected utility associated to the policy $\pi $ |

${\overrightarrow{u}}_{f}$ | fluid velocity vector (m/s) |

${\overrightarrow{u}}_{p}$ | particle velocity vector (m/s) |

${\overrightarrow{u}}_{s}$ | velocity of control element of a portion of string (m/s) |

$V$ | volume (m^{3}) |

${V}_{p}$ | particle volume (m^{3}) |

$v$ | slip velocity between the two surfaces (m/s) |

${v}_{cs}$ | critical Stribeck velocity (m/s) |

$w$ | deflection in a perpendicular direction to $\overrightarrow{t}$ (m) |

Greek letters | |

α | volumetric coefficient of thermal expansion (K^{−1}) |

$\epsilon $ | Gaussian white noise |

$\lambda $ | quality coefficient for the friction estimation process |

${\rho}_{f}$ | density of the background fluid (kg/m^{3}) |

${\rho}_{m}$ | drilling fluid density (kg/m^{3}) |

${\rho}_{p}$ | density of particle (kg/m^{3}) |

${\rho}_{s}$ | density of solid constituting the string (kg/m^{3}) |

$\pi $ | a policy |

${\pi}^{*}$ | optimum policy |

$\mu $ | mass per unit length (kg/m) |

${\mu}_{a}$ | real hydraulic annulus friction factor (dimensionless) |

$\overline{{\mu}_{a}}$ | estimated hydraulic annulus friction factor (dimensionless) |

${\mu}_{k}$ | kinetic coefficient of friction (dimensionless) |

${\mu}_{r}$ | real rotational friction factor (dimensionless) |

$\overline{{\mu}_{r}}$ | estimated rotational friction factor (dimensionless) |

${\mu}_{s}$ | real sliding friction factor (dimensionless) |

$\overline{{\mu}_{s}}$ | estimated sliding friction factor (dimensionless) |

${\mu}_{t}$ | a real friction at time $t$ (dimensionless) |

$\overline{{\mu}_{t}}$ | an estimated friction at time $t$ (dimensionless) |

${\mu}_{\sigma}$ | static coefficient of friction (dimensionless) |

$\stackrel{=}{\tau}$ | stress tensor (Pa) |

${\overrightarrow{\omega}}_{p}$ | angular velocity of the particle (rd/s) |

${\overrightarrow{\omega}}_{s}$ | angular velocity of a control element of a portion of a string (rd/s) |

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**Figure 1.**Schematic representation of the modelling of the physical state of the drilling process (modified version of Figure 146 in [23]).

**Figure 2.**Schematic representation of the calibration of the friction factors (modified version of Figure 151 in [23]).

**Figure 3.**Schematic representation of the different layers of protection at the drilling process level and at the drilling machine level.

**Figure 6.**Graphical representation of a Markov Decision Process. The combination of an action ${a}_{i}$ and a state ${s}_{i}$ results in a new state ${s}_{i+1}$ and a cost ${C}_{i}$. The objective is to find an optimum policy ${\pi}^{*}$ that minimizes the accumulated cost when taking successive actions from an initial state. The transition from ${s}_{i}$ to ${s}_{i+1}$ is probabilistic, such that one has to account for the uncertainties associated to the system’s evolution.

**Figure 7.**Graphical representation of a state. The bar on the left corresponds to the depth discretization used to evaluate the drilling of the entire section. The state analysis is performed with a high depth resolution at the start, and lower resolution afterwards. The other bars show the different frictions: sliding (left), rotation (middle), annulus (right) with the downhole ones on the bottom bars and the estimated ones (for the safety triggers) on the top row. The radar plot on the right is a condensed representation: the three axes correspond to the three friction types, the black points to the estimated ones and the colored points to the downhole one. In the case where the downhole friction is higher than the estimated one, a red line is drawn between the two points. The line is green otherwise (not shown in this example).

**Figure 8.**The transitions statistically determined by simulations. The effects of each action (drilling with different speed reductions, reciprocation, friction test, and connection) on the three main frictions are recorded during the simulations. The box plots show the average values (box center) $\pm $ one (box extremities) or two (line extremities) standard deviations of the increments/decrements in frictions)

**Figure 9.**Risks associated to an action applied to a state. Given a state (top-left quadrant) and an action (in this case performing one reciprocation, one friction test and then drilling with minimum speed) one computes a cost. The components of this cost are shown in the bottom chart: to each of the successive actions is associated both an execution cost (blue part) and some risk cost. One sees that while reciprocating and performing the friction test, the risks of having an over-torque safety trigger (red) are non-neglectable, as well as the risk of being rotationally stuck (dark blue). Only this risk subsists during drilling, since the top-drive safety trigger is not active then.

**Figure 10.**A drilling plan. The top graph shows the block position evolution corresponding to a sequence of actions. In this example, the 4 first stands are drilled with some pace reduction, while the three last ones also include some reciprocation sequence.

**Figure 11.**An alternative drilling plan to the one of Figure 10. The state reached after the 6th stand differs. Then, the remaining actions suggested by the system are also affected.

**Figure 12.**The two initial states used to generate the path shown in Figure 13. The only difference is the downhole sliding friction (yellow dot), larger than its estimated counterpart in the first state, and equal to it in the second one.

**Figure 13.**The action plans as suggested by the system for the two initial conditions from Figure 12. In the first case, a reciprocation sequence is recommended, while it is not deemed necessary in the second case.

**Figure 14.**Effect of uncertainty on the decision-making. The uncertainty in friction estimation is high in the low plots (the standard deviations are indicated by the green markers). In this situation, the system recommends performing a friction test in addition to the reciprocation sequence.

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## Share and Cite

**MDPI and ACS Style**

Cayeux, E.; Daireaux, B.; Ambrus, A.; Mihai, R.; Carlsen, L.
Autonomous Decision-Making While Drilling. *Energies* **2021**, *14*, 969.
https://doi.org/10.3390/en14040969

**AMA Style**

Cayeux E, Daireaux B, Ambrus A, Mihai R, Carlsen L.
Autonomous Decision-Making While Drilling. *Energies*. 2021; 14(4):969.
https://doi.org/10.3390/en14040969

**Chicago/Turabian Style**

Cayeux, Eric, Benoît Daireaux, Adrian Ambrus, Rodica Mihai, and Liv Carlsen.
2021. "Autonomous Decision-Making While Drilling" *Energies* 14, no. 4: 969.
https://doi.org/10.3390/en14040969