Parameters Estimation and Prediction of Water Movement and Solute Transport in Layered, Variably Saturated Soils Using the Ensemble Kalman Filter

The parameters of water movement and solute transport models are essential for the accurate simulation of soil moisture and salinity, particularly for layered soils in field conditions. Parameter estimation can be achieved using the inverse modeling method. However, this type of method cannot fully consider the uncertainties of measurements, boundary conditions, and parameters, resulting in inaccurate estimations of parameters and predictions of state variables. The ensemble Kalman filter (EnKF) is well-suited to data assimilation and parameter prediction in Situations with large numbers of variables and uncertainties. Thus, in this study, the EnKF was used to estimate the parameters of water movement and solute transport in layered, variably saturated soils. Our results indicate that when used in conjunction with the HYDRUS-1D software (University of California Riverside, California, CA, USA) the EnKF effectively estimates parameters and predicts state variables for layered, variably saturated soils. The assimilation of factors such as the initial perturbation and ensemble size significantly affected in the simulated results. A proposed ensemble size range of 50–100 was used when applying the EnKF to the highly nonlinear hydrological models of the present study. Although the simulation results for moisture did not exhibit substantial improvement with the assimilation, the simulation of the salinity was significantly improved through the assimilation of the salinity and relative solutetransport parameters. Reducing the uncertainties in measured data can improve the goodness-of-fit in the application of the EnKF method. Sparse field condition observation data also benefited from the accurate measurement of state variables in the case of EnKF assimilation. However, the application of the EnKF algorithm for layered, variably saturated soils with hydrological models requires further study, because it is a challenging and highly nonlinear problem.

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An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation

SPE Journal ◽

10.2118/108438-pa ◽

2007 ◽

Vol 12 (04) ◽

pp. 438-446 ◽

Cited By ~ 172

Author(s):

Yaqing Gu ◽

Dean S. Oliver

Keyword(s):

Porous Media ◽

Kalman Filter ◽

Data Assimilation ◽

Ensemble Kalman Filter ◽

Weather Forecasting ◽

Weather Prediction ◽

Flow In Porous Media ◽

Petroleum Reservoir ◽

State Variables ◽

Highly Nonlinear

Summary The dynamical equations for multiphase flow in porous media are highly nonlinear and the number of variables required to characterize the medium is usually large, often two or more variables per simulator gridblock. Neither the extended Kalman filter nor the ensemble Kalman filter is suitable for assimilating data or for characterizing uncertainty for this type of problem. Although the ensemble Kalman filter handles the nonlinear dynamics correctly during the forecast step, it sometimes fails badly in the analysis (or updating) of saturations. This paper focuses on the use of an iterative ensemble Kalman filter for data assimilation in nonlinear problems, especially of the type related to multiphase ow in porous media. Two issues are key:iteration to enforce constraints andensuring that the resulting ensemble is representative of the conditional pdf (i.e., that the uncertainty quantification is correct). The new algorithm is compared to the ensemble Kalman filter on several highly nonlinear example problems, and shown to be superior in the prediction of uncertainty. Introduction For linear problems, the Kalman filter is optimal for assimilating measurements to continuously update the estimate of state variables. Kalman filters have occasionally been applied to the problem of estimating values of petroleum reservoir variables (Eisenmann et al. 1994; Corser et al. 2000), but they are most appropriate when the problems are characterized by a small number of variables and when the variables to be estimated are linearly related to the observations. Most data assimilation problems in petroleum reservoir engineering are highly nonlinear and are characterized by many variables, often two or more variables per simulator gridblock. The problem of weather forecasting is in many respects similar to the problem of predicting future petroleum reservoir performance. The economic impact of inaccurate predictions is substantial in both cases, as is the difficulty of assimilating very large data sets and updating very large numerical models. One method that has been recently developed for assimilating data in weather forecasting is ensemble Kalman filtering (Evensen 1994; Houtekamer and Mitchell 1998; Anderson and Anderson 1999; Hamill et al. 2000; Houtekamer and Mitchell 2001; Evensen 2003). It has been demonstrated to be useful for weather prediction over the North Atlantic. The method is now beginning to be applied for data assimilation in groundwater hydrology (Reichle et al. 2002; Chen and Zhang 2006) and in petroleum engineering (Nævdal et al. 2002, 2005; Gu and Oliver 2005; Liu and Oliver 2005a; Wen and Chen 2006, 2007; Zafari and Reynolds 2007; Gao et al. 2006; Lorentzen et al. 2005; Skjervheim et al. 2007; Dong et al. 2006), but the applications to state variables whose density functions are bimodal has proved problematic (Gu and Oliver 2006). For applications to nonlinear assimilation problems, it is useful to think of the ensemble Kalman filter as a least squares method that obtains an averaged gradient for minimization not from a variational approach but from an empirical correlation between model variables (Anderson 2003; Zafari et al. 2006). In addition to providing a mean estimate of the variables, a Monte Carlo estimate of uncertainty can be obtained directly from the variability in the ensemble.

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Data Assimilation Using the Constrained Ensemble Kalman Filter

SPE Journal ◽

10.2118/125101-pa ◽

2010 ◽

Vol 16 (02) ◽

pp. 331-342 ◽

Cited By ~ 13

Author(s):

Hemant A. Phale ◽

Dean S. Oliver

Keyword(s):

Kalman Filter ◽

Data Assimilation ◽

Ensemble Kalman Filter ◽

History Matching ◽

Model Parameters ◽

State Variables ◽

Reservoir Properties ◽

Compositional Model ◽

Highly Nonlinear ◽

Nonnegativity Constraints

Summary When the ensemble Kalman filter (EnKF) is used for history matching, the resulting updates to reservoir properties sometimes exceed physical bounds, especially when the problem is highly nonlinear. Problems of this type are often encountered during history matching compositional models using the EnKF. In this paper, we illustrate the problem using an example in which the updated molar density of CO2 in some regions is observed to take negative values while molar densities of the remaining components are increased. Standard truncation schemes avoid negative values of molar densities but do not address the problem of increased molar densities of other components. The results can include a spurious increase in reservoir pressure with a subsequent inability to maintain injection. In this paper, we present a method for constrained EnKF (CEnKF), which takes into account the physical constraints on the plausible values of state variables during data assimilation. In the proposed method, inequality constraints are converted to a small number of equality constraints, which are used as virtual observations for calibrating the model parameters within plausible ranges. The CEnKF method is tested on a 2D compositional model and on a highly heterogeneous three-phase-flow reservoir model. The effect of the constraints on mass conservation is illustrated using a 1D Buckley-Leverett flow example. Results show that the CEnKF technique is able to enforce the nonnegativity constraints on molar densities and the bound constraints on saturations (all phase saturations must be between 0 and 1) and achieve a better estimation of reservoir properties than is obtained using only truncation with the EnKF.

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On the Choice of an Optimal Localization Radius in Ensemble Kalman Filter Methods

Monthly Weather Review ◽

10.1175/mwr-d-13-00246.1 ◽

2014 ◽

Vol 142 (6) ◽

pp. 2165-2175 ◽

Cited By ~ 24

Author(s):

Paul Kirchgessner ◽

Lars Nerger ◽

Angelika Bunse-Gerstner

Keyword(s):

Kalman Filter ◽

Ensemble Kalman Filter ◽

Degrees Of Freedom ◽

Localization Length ◽

Ensemble Size ◽

Localization Radius ◽

Filter Methods ◽

Local Observation ◽

Optimal Localization ◽

Analysis Quality

Abstract In data assimilation applications using ensemble Kalman filter methods, localization is necessary to make the method work with high-dimensional geophysical models. For ensemble square root Kalman filters, domain localization (DL) and observation localization (OL) are commonly used. Depending on the localization method, appropriate values have to be chosen for the localization parameters, such as the localization length and the weight function. Although frequently used, the properties of the localization techniques are not fully investigated. Thus, up to now an optimal choice for these parameters is a priori unknown and they are generally found by expensive numerical experiments. In this study, the relationship between the localization length and the ensemble size in DL and OL is studied using twin experiments with the Lorenz-96 model and a two-dimensional shallow-water model. For both models, it is found that the optimal localization length for DL and OL depends linearly on an effective local observation dimension that is given by the sum of the observation weights. In the experiments no influence of the model dynamics on the optimal localization length was observed. The effective observation dimension defines the degrees of freedom that are required for assimilating observations, while the ensemble size defines the available degrees of freedom. Setting the localization radius such that the effective local observation dimension equals the ensemble size yields an adaptive localization radius. Its performance is tested using a global ocean model. The experiments show that the analysis quality using the adaptive localization is similar to the analysis quality of an optimally tuned constant localization radius.

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Engine Parameter Estimation in Test Cells Using Hybrid Physics/Empirical Models

Volume 3: Controls, Diagnostics and Instrumentation; Education; Electric Power; Microturbines and Small Turbomachinery; Solar Brayton and Rankine Cycle ◽

10.1115/gt2011-45633 ◽

2011 ◽

Cited By ~ 1

Author(s):

Jonathan A. DeCastro ◽

Dean K. Frederick ◽

Liang Tang

Keyword(s):

Kalman Filter ◽

Hybrid Model ◽

Dynamic Performance ◽

Health Condition ◽

State Variables ◽

Engine Model ◽

Physical Plant ◽

Highly Nonlinear ◽

Test Cells ◽

Estimation System

Estimation of engine parameters such as thrust in test cells is a difficult process due to the highly nonlinear nature of the engine dynamics, the complex interdependency of thrust and the engine’s health condition, and factors that corrupt thrust measurements due to test stand construction. Because the frequency content of the corrupting dynamics is close to the engine’s dynamics, filtering the thrust signal is not sufficient for extraction of the true dynamic content. A configurable thrust estimation system is developed for accurate data reduction which provides “virtual” measurements of thrust and other necessary parameters at steady state and during aggressive engine transients. The thrust estimation framework consists of a representative nonlinear engine model coupled with an adaptive structural dynamics model. To account for discrepancies between the physics-based model and the true engine, a hybrid model using a novel neural network (NN) enhancement to a physics-based engine model is presented that reduces certain modeling errors between the engine model and the physical plant. This includes engine-to-engine variation, engine degradation and any essential neglected dynamics. To fuse the model and sensor measurements, this hybrid model is used within a constant-gain extended Kalman filter batch estimator which is able to reconstruct the true dynamic performance of the engine using noisy or corrupted sensor measurements and control inputs. The Kalman filter estimates measured and unmeasured parameters and state variables such as engine component deterioration parameters and effective flow areas.

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Optimal ensemble size of ensemble Kalman filter in sequential soil moisture data assimilation

Geophysical Research Letters ◽

10.1002/2015gl063366 ◽

2015 ◽

Vol 42 (16) ◽

pp. 6710-6715 ◽

Cited By ~ 29

Author(s):

Jifu Yin ◽

Xiwu Zhan ◽

Youfei Zheng ◽

Christopher R. Hain ◽

Jicheng Liu ◽

...

Keyword(s):

Soil Moisture ◽

Kalman Filter ◽

Data Assimilation ◽

Ensemble Kalman Filter ◽

Ensemble Size ◽

Soil Moisture Data ◽

Optimal Ensemble

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Sensitivity tests for an ensemble Kalman filter for aerosol assimilation

Atmospheric Chemistry and Physics Discussions ◽

10.5194/acpd-10-5947-2010 ◽

2010 ◽

Vol 10 (3) ◽

pp. 5947-5997

Author(s):

N. A. J. Schutgens ◽

T. Miyoshi ◽

T. Takemura ◽

T. Nakajima

Keyword(s):

Kalman Filter ◽

Ensemble Kalman Filter ◽

Patch Size ◽

Initial Conditions ◽

Grid Cell ◽

Ensemble Size ◽

Sensitivity Tests ◽

Local Patch ◽

The Impact ◽

Assimilation System

Abstract. We present sensitivity tests for a global aerosol assimilation system utilizing AERONET observations of AOT (aerosol optical thickness) and AAE (aerosol Ångström exponent). The assimilation system employs an ensemble Kalman filter which requires optimization of three numerical parameters: ensemble size nens, local patch size npatch and inflation factor ρ. In addition, experiments are performed to test the impact of various implementations of the system. For instance, we use a different prescription of the emission ensemble or a different combination of observations. The various experiments are compared against one-another and against independent AERONET andMODIS/Aqua observations. The assimilation leads to significant improvements in modelled AOT and AAE fields. Moreover remaining errors are mostly random while they are mostly systematic for an experiment without assimilation. In addition, these results do not depend much on our parameter or design choices. It appears that the value of the local patch size has by far the biggest impact on the assimilation, which has sufficiently converged for an ensemble size of nens=20. Assimilating AOT and AAE is clearly preferential to assimilating AOT at two different wavelengths. In contrast, initial conditions or a description of aerosol beyond two modes (coarse and fine) have only little effect. We also discuss the use of the ensemble spread as an error estimate of the analysed AOT and AAE fields. We show that a very common prescription of the emission ensemble (independent random modification in each grid cell) can have trouble generating sufficient spread in the forecast ensemble.

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Data assimilation methods for estimating a heterogeneous conductivity field by assimilating transient solute transport data via ensemble Kalman filter

Hydrological Processes ◽

10.1002/hyp.9523 ◽

2012 ◽

Vol 27 (26) ◽

pp. 3873-3884 ◽

Cited By ~ 11

Author(s):

Juxiu Tong ◽

Bill X. Hu ◽

Jinzhong Yang

Keyword(s):

Kalman Filter ◽

Data Assimilation ◽

Solute Transport ◽

Ensemble Kalman Filter

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State estimation of conceptual hydrological models using unscented Kalman filter

Hydrology Research ◽

10.2166/nh.2018.038 ◽

2018 ◽

Vol 50 (2) ◽

pp. 479-497 ◽

Cited By ~ 2

Author(s):

P. Jiang ◽

Y. Sun ◽

W. Bao

Keyword(s):

Kalman Filter ◽

State Estimation ◽

Unscented Kalman Filter ◽

Hydrologic Model ◽

Generation Process ◽

Hydrological Models ◽

State Variables ◽

Symmetric Point ◽

The Difference

Abstract Unscented Kalman filter (UKF) has its origin in transforming the Gaussian random variables for nonlinear estimation and has received little attention in the context of state estimation of conceptual hydrological models. This paper introduces UKF to estimate state variables of a conceptual hydrologic model. A symmetric point approach and a scaling framework are used for performing the sample generation process of UKF. This paper investigates the application of UKF for state estimation with a synthetic case study in which both the simulated state, the true state, and the corrected state are precisely known. The results show that the use of UKF can improve the performance of both the model outputs and the state variables as the difference between the corrected trajectories and the true trajectories decreases rapidly and tends to vanish after only a few iterations. Our results and comparisons also demonstrated the capability and usefulness of UKF for state estimation in two real basins.

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Use of Phase Streamlines for Covariance Localization in Ensemble Kalman Filter for Three-Phase History Matching

SPE Reservoir Evaluation & Engineering ◽

10.2118/144579-pa ◽

2012 ◽

Vol 15 (03) ◽

pp. 273-289 ◽

Cited By ~ 4

Author(s):

Shingo Watanabe ◽

Akhil Datta-Gupta

Keyword(s):

Kalman Filter ◽

Gas Phase ◽

Ensemble Kalman Filter ◽

History Matching ◽

Model Parameters ◽

Ensemble Size ◽

Three Phase ◽

Cross Covariance ◽

The Cross ◽

Water Cut

Summary The ensemble Kalman filter (EnKF) has gained increased popularity for history matching and continuous reservoir-model updating. It is a sequential Monte Carlo approach that works with an ensemble of reservoir models. Specifically, the method uses cross covariance between measurements and model parameters estimated from the ensemble. For practical field applications, the ensemble size needs to be kept small for computational efficiency. However, this leads to poor approximations of the cross covariance and can cause loss of geologic realism from unrealistic model updates outside the region of the data influence and/or loss of variance leading to ensemble collapse. A common approach to remedy the situation is to limit the influence of the data through covariance localization. In this paper, we show that for three-phase-flow conditions, the region of covariance localization strongly depends on the underlying flow dynamics as well as on the particular data type that is being assimilated, for example, water cut or gas/oil ratio (GOR). This makes the traditional distance-based localizations suboptimal and, often, ineffective. Instead, we propose the use of water- and gas-phase streamlines as a means for covariance localization for water-cut- and GOR-data assimilation. The phase streamlines can be computed on the basis of individual-phase velocities which are readily available after flow simulation. Unlike the total velocity streamlines, phase streamlines can be discontinuous. We show that the discontinuities in water-phase and gas-phase streamlines naturally define the region of influence for water-cut and GOR data and provide a flow-relevant covariance localization during EnKF updating. We first demonstrate the validity of the proposed localization approach using a waterflood example in a quarter-five-spot pattern. Specifically, we compare the phase streamline trajectories with cross-covariance maps computed using an ensemble size of 2,000 for both water-cut and GOR data. The results show a close correspondence between the time evolution of phase streamlines and the cross-covariance maps of water-cut and GOR data. A benchmark uncertainty quantification (the PUNQ-S3) (Carter 2007) model application shows that our proposed localization outperforms the distance-based localization method. The updated models show improved forecasts while preserving geologic realism.

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