Continuous learning of analytical and machine learning rate of penetration (ROP) models for real-time drilling optimization

Oil and gas operators strive to reach hydrocarbon reserves by drilling wells in the safest and fastest possible manner, providing indispensable energy to society at reduced costs while maintaining environmental sustainability. Real-time drilling optimization consists of selecting operational drilling parameters that maximize a desirable measure of drilling performance. Drilling optimization efforts often aspire to improve drilling speed, commonly referred to as rate of penetration (ROP). ROP is a function of the forces and moments applied to the bit, in addition to mud, formation, bit and hydraulic properties. Three operational drilling parameters may be constantly adjusted at surface to influence ROP towards a drilling objective: weight on bit (WOB), drillstring rotational speed (RPM), and drilling fluid (mud) flow rate. In the traditional, analytical approach to ROP modeling, inflexible equations relate WOB, RPM, flow rate and/or other measurable drilling parameters to ROP and empirical model coefficients are computed for each rock formation to best fit field data. Over the last decade, enhanced data acquisition technology and widespread cheap computational power have driven a surge in applications of machine learning (ML) techniques to ROP prediction. Machine learning algorithms leverage statistics to uncover relations between any prescribed inputs (features/predictors) and the quantity of interest (response). The biggest advantage of ML algorithms over analytical models is their flexibility in model form. With no set equation, ML models permit segmentation of the drilling operational parameter space. However, increased model complexity diminishes interpretability of how an adjustment to the inputs will affect the output. There is no single ROP model applicable in every situation. This study investigates all stages of the drilling optimization workflow, with emphasis on real-time continuous model learning. Sensors constantly record data as wells are drilled, and it is postulated that ROP models can be retrained in real-time to adapt to changing drilling conditions. Cross-validation is assessed as a methodology to select the best performing ROP model for each drilling optimization interval in real-time. Constrained to rig equipment and operational limitations, drilling parameters are optimized in intervals with the most accurate ROP model determined by cross-validation. Dynamic range and full range training data segmentation techniques contest the classical lithology-dependent approach to ROP modeling. Spatial proximity and parameter similarity sample weighting expand data partitioning capabilities during model training. The prescribed ROP modeling and drilling parameter optimization scenarios are evaluated according to model performance, ROP improvements and computational expensePetroleum and Geosystems Engineerin

A Recipe for the Estimation of Information Flow in a Dynamical System

Information-theoretic quantities, such as entropy and mutual information (MI), can be used to quantify the amount of information needed to describe a dataset or the information shared between two datasets. In the case of a dynamical system, the behavior of the relevant variables can be tightly coupled, such that information about one variable at a given instance in time may provide information about other variables at later instances in time. This is often viewed as a flow of information, and tracking such a flow can reveal relationships among the system variables. Since the MI is a symmetric quantity; an asymmetric quantity, called Transfer Entropy (TE), has been proposed to estimate the directionality of the coupling. However, accurate estimation of entropy-based measures is notoriously difficult. Every method has its own free tuning parameter(s) and there is no consensus on an optimal way of estimating the TE from a dataset. We propose a new methodology to estimate TE and apply a set of methods together as an accuracy cross-check to provide a reliable mathematical tool for any given data set. We demonstrate both the variability in TE estimation across techniques as well as the benefits of the proposed methodology to reliably estimate the directionality of coupling among variables

Generalized Duffy transformation for integrating vertex singularities

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation