Flexible iterative ensemble smoother for calibration of perfect and imperfect models

Iterative ensemble smoothers have been widely used for calibrating simulators of various physical systems due to the relatively low computational cost and the parallel nature of the algorithm. However, iterative ensemble smoothers have been designed for perfect models under the main assumption that the specified physical models and subsequent discretized mathematical models have the capability to model the reality accurately. While significant efforts are usually made to ensure the accuracy of the mathematical model, it is widely known that the physical models are only an approximation of reality. These approximations commonly introduce some type of model error which is generally unknown and when the models are calibrated, the effects of the model errors could be smeared by adjusting the model parameters to match historical observations. This results in a bias estimated parameters and as a consequence might result in predictions with questionable quality. In this paper, we formulate a flexible iterative ensemble smoother, which can be used to calibrate imperfect models where model errors cannot be neglected. We base our method on the ensemble smoother with multiple data assimilation (ES-MDA) as it is one of the most widely used iterative ensemble smoothing techniques. In the proposed algorithm, the residual (data mismatch) is split into two parts. One part is used to derive the parameter update and the second part is used to represent the model error. The proposed method is quite general and relaxes many of the assumptions commonly introduced in the literature. We observe that the proposed algorithm has the capability to reduce the effect of model bias by capturing the unknown model errors, thus improving the quality of the estimated parameters and prediction capacity of imperfect physical models.

Joint Estimation of Hydraulic and Biochemical Parameters for Reactive Transport Modelling with a Modified ILUES Algorithm

Multicomponent reactive transport modeling is a powerful tool for the comprehensive analysis of coupled hydraulic and biochemical processes. The performance of the simulation model depends on the accuracy of related model parameters whose values are usually difficult to determine from direct measurements. In this situation, estimates of these uncertain parameters can be obtained by solving inverse problems. In this study, an efficient data assimilation method, the iterative local updating ensemble smoother (ILUES), is employed for the joint estimation of hydraulic parameters, biochemical parameters and contaminant source characteristics in the sequential biodegradation process of tetrachloroethene (PCE). In the framework of the ILUES algorithm, parameter estimation is realized by updating local ensemble with the iterative ensemble smoother (IES). To better explore the parameter space, the original ILUES algorithm is modified by determining the local ensemble partly with a linear ranking selection scheme. Numerical case studies based on the sequential biodegradation of PCE are then used to evaluate the performance of the ILUES algorithm. The results show that the ILUES algorithm is able to achieve an accurate joint estimation of related model parameters in the reactive transport model.

Data assimilation of subsurface flow via iterative ensemble smoother and physics-informed neural network

<p>Subsurface flow problems usually involve some degree of uncertainty. For reducing the uncertainty of subsurface flow prediction, data assimilation is usually necessary. Data assimilation is time consuming. In order to improve the efficiency of data assimilation, surrogate model of subsurface flow problem may be utilized. In this work, a physics-informed neural network (PINN) based surrogate model is proposed for subsurface flow with uncertain model parameters. Training data generated by solving stochastic partial differential equations (SPDEs) are utilized to train the neural network. Besides the data mismatch term, the term that incorporates physics laws is added in the loss function. The trained neural network can predict the solutions of the subsurface flow problem with new stochastic parameters, which can serve as a surrogate for approximating the relationship between model output and model input. By incorporating physics laws, PINN can achieve high accuracy. Then an iterative ensemble smoother (ES) is introduced to implement the data assimilation task based on the PINN surrogate. Several subsurface flow cases are designed to test the performance of the proposed paradigm. The results show that the PINN surrogate can significantly improve the efficiency of data assimilation task while maintaining a high accuracy.</p>

Improved Estimation and Forecasting Through Residual-Based Model Error Quantification

Summary Owing to the complex nature of hydrocarbon reservoirs, the numerical model constructed by geoscientists is always a simplified version of reality: for example, it might lack resolution from discretization and lack accuracy in modeling some physical processes. This flaw in the model that causes mismatch between actual observations and simulated data when “perfect” model parameters are used as model inputs is known as “model error”. Even in a situation when the model is a perfect representation of reality, the inputs to the model are never completely known. During a typical model calibration procedure, only a subset of model inputs is adjusted to improve the agreement between model responses and historical data. The remaining model inputs that are not calibrated and are likely fixed at incorrect values result in model error in a similar manner as the imperfect model scenario. Assimilation of data without accounting for model error can result in the incorrect adjustment to model parameters, the underestimation of prediction uncertainties, and bias in forecasts. In this paper, we investigate the benefit of recognizing and accounting for model error when an iterative ensemble smoother is used to assimilate production data. The correlated “total error” (a combination of model error and observation error) is estimated from the data residual after a standard history-matching using the Levenberg-Marquardt form of iterative ensemble smoother (LM-EnRML). This total error is then used in further data assimilations to improve the estimation of model parameters and quantification of prediction uncertainty. We first illustrate the method using a synthetic 2D five-spot example, where some model errors are deliberately introduced, and the results are closely examined against the known “true” model. Then, the Norne field case is used to further evaluate the method. The Norne model has previously been history-matched using the LM-EnRML (Chen and Oliver 2014), where cell-by-cell properties (permeability, porosity, net-to-gross, vertical transmissibility) and parameters related to fault transmissibility, depths of water/oil contacts, and relative permeability function are adjusted to honor historical data. In this previous study, the authors highlighted the importance of including large amounts of model parameters, the proper use of localization, and heuristic adjustment of data noise to account for modeling error. In this paper, we improve the last aspect by quantitatively estimating model error using residual analysis.

Ensemble-based seismic inversion for a stratified medium

Seismic waveform inversion is a nontrivial optimization task, which is often complicated by the nonlinear relationship between the elastic attributes of interest and the large amount of data obtained in seismic experiments. Quantifying the solution uncertainty can be even more challenging, and it requires considering the problem in a probabilistic setting. Consequently, the seismic inverse problem is placed in a Bayesian framework, using a sequential filtering approach to invert for the elastic parameters. The method uses an iterative ensemble smoother to estimate the subsurface parameters, and from the ensemble, a notion of estimation uncertainty is readily available. The ensemble implicitly linearizes the relation between the parameters and the observed waveform data; hence, it requires no tangent linear model. The approach is based on sequential conditioning on partitions of the whole data record (1) to regularize the inversion path and effectively drive the estimation process in a top-down manner and (2) to circumvent a consequence of the ensemble reduced rank approximation. The method is exemplified on a synthetic case, inverting for elastic parameters in a 1D medium using a seismic shot record. Our results indicate that the iterative ensemble method is applicable to seismic waveform inversion and that the ensemble representation indeed indicates estimation uncertainty.