Application of ensemble transform data assimilation methods for parameter estimation in reservoir modelling

Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable methods of MCMC are computationally expensive. Sequential ensemble methods such as ensemble Kalman filers and particle filters provide a favourable alternative. However, Ensemble Kalman Filter has an assumption of Gaussianity. Ensemble Transform Particle Filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimation in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ Ensemble Transform Particle Filter (ETPF) and Ensemble Transform Kalman Filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup and observe that for a small number of uncertain parameters (1 and 5) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands a knowledge at which mode to truncate

Accounting for model error in Tempered Ensemble Transform Particle Filter and its application to non-additive model error

In this paper, we trivially extend Tempered (Localized) Ensemble Transform Particle Filter---T(L)ETPF---to account for model error. We examine T(L)ETPF performance for non-additive model error in a low-dimensional and a high-dimensional test problem. The former one is a nonlinear toy model, where uncertain parameters are non-Gaussian distributed but model error is Gaussian distributed. The latter one is a steady-state single-phase Darcy flow model, where uncertain parameters are Gaussian distributed but model error is non-Gaussian distributed. The source of model error in the Darcy flow problem is uncertain boundary conditions. We comapare T(L)ETPF to a Regularized (Localized) Ensemble Kalman Filter---R(L)EnKF. We show that T(L)ETPF outperforms R(L)EnKF for both the low-dimensional and the high-dimensional problem. This holds even when ensemble size of TLETPF is 100 while ensemble size of R(L)EnKF is greater than 6000. As a side note, we show that TLETPF takes less iterations than TETPF, which decreases computational costs; while RLEnKF takes more iterations than REnKF, which incerases computational costs. This is due to an influence of localization on a tempering and a regularizing parameter

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI

Parameter estimation for subsurface flow using ensemble data assimilation

Over the years, different data assimilation methods have been implemented to acquire improved estimations of model parameters by adjusting the uncertain parameter values in such a way that the mathematical model approximates the observed data as closely and consistently as possible. However, most of these methods are developed on the assumption of Gaussianity, e.g. Ensemble Kalman Filters, whic

Accounting for model error in Tempered Ensemble Transform Particle Filter and its application to non-additive model error

In this paper, we trivially extend Tempered (Localized) Ensemble Transform Particle Filter—T(L)ETPF—to account for model error. We examine T(L)ETPF performance for non-additive model error in a low-dimensional and a high-dimensional test problem. The former one is a nonlinear toy model, where uncertain parameters are non-Gaussian distributed but model error is Gaussian distributed. The latter one is

Comparison of regularized ensemble Kalman filter and tempered ensemble transform particle filter for an elliptic inverse problem with uncertain boundary conditions

In this paper, we focus on parameter estimation for an elliptic inverse problem. We consider a 2D steady-state single- phase Darcy flow model, where permeability and boundary conditions are uncertain. Permeability is parameterized by the Karhunen-Loeve expansion and thus assumed to be Gaussian distributed. We employ two ensemble-based data assimilation methods: ensemble Kalman filter and ensemble transf

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high- dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI

Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation

Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark

Parameter Estimation in Random Energy Systems using Data Assimilation

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