a b s t r a c tDomain decomposition (DD) methods, such as the additive Schwarz method, are almost exclusively applied to linearized equations. In the context of nonlinear problems, these linear systems appear as part of a Newton iteration. However, applying DD methods directly to the original nonlinear problem has some attractive features, most notably that the Newton iterations now solve local problems, and thus are expected to be simpler. Furthermore, strong, local nonlinearities may to a less extent affect the numerical algorithm. For linear problems, DD can be applied both as an iterative solver or as a preconditioner. For nonlinear problems, it has until recently only been understood how to use DD as a solver.This article offers a systematic study of domain decomposition strategies in the context of nonlinear porous-medium flow problems. The study thus compares four different approaches, which represents DD applied both as a solver and preconditioner, to both the linearized and nonlinear equations. Our model equations are those obtained from a fully implicit discretization of immiscible two-phase flow in heterogeneous porous media. In particular we emphasize the case of nonlinear preconditioning, an algorithm that to our knowledge so far has not been studied nor implemented for flow in porous media. Our results show that the novel algorithm is up to 75% faster than the standard algorithm for the most challenging problems for a moderate number of subdomains.

Solutions of a boundary value problem for the Korteweg-de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient.

Solving realistic problems related to flow in porous media to desired accuracy may be prohibitively expensive with available computing resources. Multiscale effects and nonlinearities in the governing equations are among the most important contributors to this situation. Hence, developing methods that handle these features better is essential in order to be able to solve the problems more efficiently. Focus has until recently largely been on preconditioners for linearized problems. This article proposes a two-scale nonlinear preconditioning technique for flow problems in porous media that allows for incorporating physical intuition directly in the preconditioner. By assuming a certain dominant physical process, this technique will resemble upscaling in the equilibrium limit, with the computational benefits that follow. In this study, the method is established as a preconditioner with good scalability properties for challenging problems regardless of dominant physics, thus laying the foundation for further studies with physical information in the preconditioner.

Monitoring of Equivalent Circulating Density (ECD) may improve assessment of potential bad hole cleaning conditions if calculated and measured sufficiently accurately. Machine learning (ML) models can be used for predicting ECD integrating both along-string and surface drilling measurements and physics-based model (PBM) results, even though their generalization is often challenging. To remediate this generalizability issue, we present an adaptative predictive deep-learning model that is retrained with new measurements in real-time, conditionally that the new measurements are not detected as anomalies. Past ECD measurements, corresponding values predicted by a 1D PBM and other drilling measurements are used as input to a deep learning model, which is pretrained on historical drilling data without any hole cleaning problem. This model has two components: an anomaly detector, and a predictor. In this paper, both components are based on combinations of Long Short-Term Memory (LSTM) cells that allow (1) to account for data correlations between the different time series and between the different time stamps, and (2) generate future data conditioned to past observations. As drilling progresses, new data is proposed to the anomaly detector: if the network fails to reconstruct them correctly, an alarm is raised. Otherwise, the new data is used to retrain the models. We show the benefits of such an approach on two real examples from offshore Norway with increasing complexity: For the first one, with no major drilling issue, we simply use ECD from the PBM to predict ECD ahead of the bit. The second example had multiple issues linked with mud loss and poor hole cleaning. For this latter case, we used additional topside measurements to better constrain the ECD prediction.