Parameter Estimation in Random Energy Systems using Data Assimilation

Abstract

With the increasing energy demand and limited natural fuel reserves, efficient utilization of oil/gas reservoirs is of paramount importance. An oil/gas reservoir represents a natural accumulation of hydrocarbons within different rock structures. Hence, an optimal oil enhanced strategy inevitably depends on the petrophysical properties of such rock structures. By knowing the lithological structure and petrophysical properties, a so-called forward model can be solved to predict the reservoir performance. However, most reservoirs are buried hundreds of meters underneath the earth's surface, and thus makes direct measurements of the rock properties extremely difficult. Usually a prior information about the geophysical properties is given, which still needs to be corrected by indirect measurements. These measurements are, however, known only at the production well locations that are often hundreds of meters apart and corrupted by errors. This poses an ill-posed inverse problem of estimating uncertain rock properties from the measurements, since many possible combinations of uncertain properties can result in equally good matches to the measurements. In meteorology, Bayesian framework based data assimilation is a widely-used mathematical methodology which combines measurements of meteorological variables with computational weather models and gives state prediction by correcting the initial conditions. The recent advancement in meteorology is aimed at fully Bayesian computational methods (e.g. particle filtering) to minimize the effect of erroneous assumptions and approximations. Unlike meteorology, where uncertain fields evolve over time, in geoscience the rock properties are stationary. This thesis investigates the application of particle filtering methods to inverse problems. Furthermore, this thesis adopts ensemble transform particle filtering method (ETPF), and introduces a novel approach, tempered ensemble transform particle filter (TETPF), to estimate the petrophysical parameters of reservoir model. For this purpose, we undertake a test case of steady-state single-phase Darcy flow model with permeability as an uncertain parameter. We examine the performance of the data assimilation methods in a twin experiment setup, where the observations of pressure are synthetically created from the same model but based on different values of permeability. The numerical experiments demonstrate that the ETPF gives a good estimation when the number of uncertain parameters is small and is Gaussian-distributed. However, it struggles when facing a large number of uncertain parameters and requires localization. The localized versions of data assimilation method reduces the degrees of freedom and updates uncertain grid-dependent parameters locally. The introduced method, TETPF, outperforms the other particle filtering approaches for a high-dimensional problem with non-Gaussian distributed parameters. This has high relevance for the real subsurface reservoir, as it allows for complex structures that include channels with different types of rocks. TETPF requires smaller ensemble sizes compared to the standard particle filtering approaches and provides improved posterior distribution of the non-Gaussian distributed parameters. Supplementing the introduced method with localization further improves the performance. Though the novel data assimilation method requires a computationally affordable ensemble of reservoir models, by itself it remains computationally expensive. The future development focuses on investigation of computationally cheaper extensions of the introduced method and its application on more realistic scenarios.