Bayesian change point prediction for downhole drilling pressures with hidden Markov models

In the drilling of oil wells, the need to accurately detect downhole formation pressure transitions has long been established as critical for safety and economics. In this article, we examine the application of Hidden Markov Models (HMMs) to oilwell drilling processes with a focus on the real time evolution of downhole formation pressures in its partially observed state. The downhole drilling pressure system can be viewed as a nonlinear, non-degrading stochastic process whose optimum performance is in a region in its warning state prior to random failure in time. The differential pressure system $\left(∆P\right)$ is modeled as a hidden 3 state continuous time Markov process. States 0 and 1 are not observable and represent the normally pressured (initiating $∆P$) and abnormally pressured or warning (reducing $∆P$) states respectively. State 2 is the observable failure state (from negative $∆P$ and loss of well control). The signal process of the evolution of differential pressure $\left(∆P\right)$ is identified in the changes in the observable rate of penetration (ROP) encoded in drilling performance data. The state and observation parameters of the HMM are estimated using the Expectation Maximization (EM) algorithm and we show, for a univariate system with a depth dependent time relationship, that the model parameter updates of the EM algorithm equation have explicit solutions. A Bayesian inference model, to determine the safety threshold of the system and early failure prediction at each sampling epoch, is thereafter proposed. The application of our stochastic model of the dynamic evolution of downhole pressures in operational time is illustrated with a hindcast case example. The analysis showed strong early indication of probable failure in real time and was validated in the field post drilling system failure that resulted in significant recovery costs. The potential to predict the differential pressure state transitions ahead of the bit represents a capability not currently available in the industry. This opens up significant opportunity for value creation from optimizing drilling operations to deliver substantial savings in well construction costs.