2,865 research outputs found
A Bayesian Consistent Dual Ensemble Kalman Filter for State-Parameter Estimation in Subsurface Hydrology
Ensemble Kalman filtering (EnKF) is an efficient approach to addressing
uncertainties in subsurface groundwater models. The EnKF sequentially
integrates field data into simulation models to obtain a better
characterization of the model's state and parameters. These are generally
estimated following joint and dual filtering strategies, in which, at each
assimilation cycle, a forecast step by the model is followed by an update step
with incoming observations. The Joint-EnKF directly updates the augmented
state-parameter vector while the Dual-EnKF employs two separate filters, first
estimating the parameters and then estimating the state based on the updated
parameters. In this paper, we reverse the order of the forecast-update steps
following the one-step-ahead (OSA) smoothing formulation of the Bayesian
filtering problem, based on which we propose a new dual EnKF scheme, the
Dual-EnKF. Compared to the Dual-EnKF, this introduces a new update
step to the state in a fully consistent Bayesian framework, which is shown to
enhance the performance of the dual filtering approach without any significant
increase in the computational cost. Numerical experiments are conducted with a
two-dimensional synthetic groundwater aquifer model to assess the performance
and robustness of the proposed Dual-EnKF, and to evaluate its
results against those of the Joint- and Dual-EnKFs. The proposed scheme is able
to successfully recover both the hydraulic head and the aquifer conductivity,
further providing reliable estimates of their uncertainties. Compared with the
standard Joint- and Dual-EnKFs, the proposed scheme is found more robust to
different assimilation settings, such as the spatial and temporal distribution
of the observations, and the level of noise in the data. Based on our
experimental setups, it yields up to 25% more accurate state and parameters
estimates
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Imaging of a fluid injection process using geophysical data - A didactic example
In many subsurface industrial applications, fluids are injected into or withdrawn from a geologic formation. It is of practical interest to quantify precisely where, when, and by how much the injected fluid alters the state of the subsurface. Routine geophysical monitoring of such processes attempts to image the way that geophysical properties, such as seismic velocities or electrical conductivity, change through time and space and to then make qualitative inferences as to where the injected fluid has migrated. The more rigorous formulation of the time-lapse geophysical inverse problem forecasts how the subsurface evolves during the course of a fluid-injection application. Using time-lapse geophysical signals as the data to be matched, the model unknowns to be estimated are the multiphysics forward-modeling parameters controlling the fluid-injection process. Properly reproducing the geophysical signature of the flow process, subsequent simulations can predict the fluid migration and alteration in the subsurface. The dynamic nature of fluid-injection processes renders imaging problems more complex than conventional geophysical imaging for static targets. This work intents to clarify the related hydrogeophysical parameter estimation concepts
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
Estimating model evidence using data assimilation
We review the field of data assimilation (DA) from a Bayesian perspective and show that, in addition to its by now common application to state estimation, DA may be used for model selection. An important special case of the latter is the discrimination between a factual modelāwhich corresponds, to the best of the modeller's knowledge, to the situation in the actual world in which a sequence of events has occurredāand a counterfactual model, in which a particular forcing or process might be absent or just quantitatively different from the actual world. Three different ensembleāDA methods are reviewed for this purpose: the ensemble Kalman filter (EnKF), the ensemble fourādimensional variational smoother (Enā4DāVar), and the iterative ensemble Kalman smoother (IEnKS). An original contextual formulation of model evidence (CME) is introduced. It is shown how to apply these three methods to compute CME, using the approximated timeādependent probability distribution functions (pdfs) each of them provide in the process of state estimation. The theoretical formulae so derived are applied to two simplified nonlinear and chaotic models: (i) the Lorenz threeāvariable convection model (L63), and (ii) the Lorenz 40āvariable midlatitude atmospheric dynamics model (L95). The numerical results of these three DAābased methods and those of an integration based on importance sampling are compared. It is found that better CME estimates are obtained by using DA, and the IEnKS method appears to be best among the DA methods. Differences among the performance of the three DAābased methods are discussed as a function of model properties. Finally, the methodology is implemented for parameter estimation and for event attribution
4D Seismic History Matching Incorporating Unsupervised Learning
The work discussed and presented in this paper focuses on the history
matching of reservoirs by integrating 4D seismic data into the inversion
process using machine learning techniques. A new integrated scheme for the
reconstruction of petrophysical properties with a modified Ensemble Smoother
with Multiple Data Assimilation (ES-MDA) in a synthetic reservoir is proposed.
The permeability field inside the reservoir is parametrised with an
unsupervised learning approach, namely K-means with Singular Value
Decomposition (K-SVD). This is combined with the Orthogonal Matching Pursuit
(OMP) technique which is very typical for sparsity promoting regularisation
schemes. Moreover, seismic attributes, in particular, acoustic impedance, are
parametrised with the Discrete Cosine Transform (DCT). This novel combination
of techniques from machine learning, sparsity regularisation, seismic imaging
and history matching aims to address the ill-posedness of the inversion of
historical production data efficiently using ES-MDA. In the numerical
experiments provided, I demonstrate that these sparse representations of the
petrophysical properties and the seismic attributes enables to obtain better
production data matches to the true production data and to quantify the
propagating waterfront better compared to more traditional methods that do not
use comparable parametrisation techniques
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