Subsurface modeling with functional data [electronic resource]
Keywords: Energy Resources Engineering.

Unconventional shale reservoirs are currently one of the most developed energy resources in the world. In the US alone, more than 1.1 million shale wells are currently in production and the estimated US shale oil reserves are at around 4.29 trillion barrels. The development of shale reservoirs is very rapid in the US. Some companies drill more than 400 horizontal shale wells per year, and this trend is likely to increase in the coming years. In such setting, quick uncertainty quantification and forecasting is of paramount importance. Conventional approaches to uncertainty quantification and forecasting were mostly found impractical in shales due poorly understood production mechanisms, high temporal requirements of uncertainty quantification studies and also because of the poor collection of scientific data. Shale reservoirs are mostly developed by small to mid size companies that usually cannot afford the collection of highly sophisticated scientific data or advanced geomodeling and simulation softwares. Moreover, transport in shales and upscaling of shale reservoir properties are very active research areas without well established practical work-flows and software. For these reasons, many reservoir modelers have adopted the so called data-driven approaches for reservoir data analysis and production forecasting. Commonly employed data-driven methodologies include regression methods for the analysis of production data and neural network based models for production forecasting. What most of the approaches employed up to date have in common is that they work with scalar outputs (i.e. 3, 6, 9 months of cumulative production or peak oil/gas), spatial correlations and spatial trends are rarely analyzed, and many of the commonly employed approaches are incapable of properly quantifying uncertainty in forecasts. In this dissertation, we take a different approach to data-driven reservoir data analysis and forecasting. Firstly, we start from a perspective that shale production profiles need to be analyzed as a whole, or as curves, and not as some discretized components of cumulative production. Secondly, we embrace the existence of spatial correlations between production profiles. It is quite intuitive that similarly completed wells would produce similar amounts of hydrocarbons if drilled reasonably close (without interference). The question is at what distance does this "similarity" disappear and how does that affect forecasts, recoverable reserve estimation and ultimate decision making? Here, we develop methodologies for interpretation, forecasting and uncertainty quantification of spatially correlated reservoir production curves or functions. The methodologies are based on the tools of relatively recently established statistical discipline called functional data analysis (FDA) and the advances in its sub-discipline, geostatistics for functional data. We demonstrate that the well known geostatististical tools such as variograms and sequential Gaussian simulation can be efficiently used for shale reservoir data analysis and forecasting. The developed methodologies are demonstrated in two unconventional reservoir case studies. The first case study analyzed gas production from ~900 horizontal wells completed in the Barnett shale, while the second case study used Anadarko Petroleum Company (APC) provided dataset with 189 wells that produced oil and gas. The previously mentioned methodologies are capable of analyzing and forecasting single variate functional data (i.e. oil rates curves only). However, the APC dataset contained wells that produced multivariate functional data, oil and gas rates over time. When analyzing such dataset the question of multivariate functional data analysis and forecasting naturally arises. The encounter of this question motivated the development of a methodology capable of analyzing and forecasting multivariate functional data. The developed methodology is based on regression trees, a well known machine learning technique, and it represents a contribution to both fields of Earth sciences and functional data analysis. The methodology is also demonstrated on the APC dataset. Functional data is not only observed in unconventional reservoir data analysis and forecasting. One also encounters functional data in conventional reservoir modeling. For example, flow simulation curves computed with conventional reservoir simulators also represent functional data. Proper numerical uncertainty quantification requires consideration of a large number of modeling parameters with wide ranges. Exhaustive exploration of such high dimensional spaces is computationally demanding and rarely achievable in practice. For this reason, modelers often employ statistical emulators that aim to interpolate or emulate the reservoir simulation solution at unexplored portions of the input space. Statistical emulators require a certain number of training runs computed with high fidelity reservoir simulators and in most applications up to date work with scalar outputs (i.e. EUR). Since reservoir simulator outputs are functional in nature, one can develop statistical emulators with the aforementioned methodologies we developed for shale reservoir forecasting. We explore this application in the last, sixth chapter of this dissertation. Another problem in conventional uncertainty quantification studies is with the use of proxy models. Proxy models are numerical models of lower fidelity and high speed that are commonly used to quickly explore the high dimensional input spaces. Their stand-alone solution is often considered noisy and sub-optimal. For this reason, modelers often employ machine learning to model the discrepancies (errors) between the proxies and their high fidelity counterparts, or they employ co-kriging based schemes that fit a statistical emulator that aggregates both proxy and high fidelity solutions in estimating unevaluated high fidelity solutions. In chapter 6, we develop novel functional co-kriging methodologies for building statistical emulators that aggregate functional responses produced by proxies and high fidelity numerical models. The methodologies are applied and compared in three case studies along with the emulators constructed with the techniques we previously used in shale reservoir modeling.