Ensemble Smoother Research Articles

An international initiative of predicting the SARS-CoV-2 pandemic using ensemble data assimilation

<p style='text-indent:20px;'>This work demonstrates the efficiency of using iterative ensemble smoothers to estimate the parameters of an SEIR model. We have extended a standard SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition the model on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters yielding model predictions in close agreement with the observations. The updated ensemble of model simulations has predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development assuming knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries, i.e., the four European countries Norway, England, The Netherlands, and France; the province of Quebec in Canada; the South American countries Argentina and Brazil; and the four US states Alabama, North Carolina, California, and New York. These countries and states all have vastly different developments of the epidemic, and we could accurately model the SARS-CoV-2 outbreak in all of them. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.</p>

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.

Using Deep Learning to Improve Ensemble Smoother: Applications to Subsurface Characterization

AbstractEnsemble smoother (ES) has been widely used in various research fields to reduce the uncertainty of the system‐of‐interest. However, the commonly adopted ES method that employs the Kalman formula, that is, ES(K), does not perform well when the probability distributions involved are non‐Gaussian. To address this issue, we suggest to use deep learning (DL) to derive an alternative analysis scheme for ES in non‐Gaussian data assimilation problems. Here we show that the DL‐based ES method, that is, ES(DL), is more general and flexible. In this new scheme, a high volume of training data is generated from a relatively small‐sized ensemble of model parameters and simulation outputs, and possible non‐Gaussian features can be preserved in the training data and captured by an adequate DL model. This new variant of ES is tested in two subsurface characterization problems with or without the Gaussian assumption. Results indicate that ES(DL) can produce similar (in the Gaussian case) or even better (in the non‐Gaussian case) results compared to those from ES(K). The success of ES(DL) comes from the power of DL in extracting complex (including non‐Gaussian) features and learning nonlinear relationships from massive amounts of training data. Although in this work we only apply the ES(DL) method in parameter estimation problems, the proposed idea can be conveniently extended to analysis of model structural uncertainty and state estimation in real‐time forecasting problems.

Recent developments combining ensemble smoother and deep generative networks for facies history matching

Ensemble smoothers are among the most successful and efficient techniques currently available for history matching. However, because these methods rely on Gaussian assumptions, their performance is severely degraded when the prior geology is described in terms of complex facies distributions. Inspired by the impressive results obtained by deep generative networks in areas such as image and video generation, we started an investigation focused on the use of autoencoders to construct a continuous parameterization for facies models. In our previous publication, we combined a convolutional variational autoencoder (VAE) with the ensemble smoother with multiple data assimilation (ES-MDA) for history matching production data in models generated with multiple-point geostatistics. Despite the good results reported in our previous publication, a major limitation of the designed parameterization is the fact that it does not allow applying distance-based localization during the ensemble smoother update, which limits its application in large-scale problems. The present work is a continuation of this research project focusing on two aspects: firstly, we benchmark nine different formulations, including VAE, generative adversarial network (GAN), Wasserstein GAN (WGAN), WGAN with gradient penalty, WGAN with spectral normalization, variational auto-encoding GAN, principal component analysis (PCA) with cycle GAN, PCA with transfer style network, and VAE with style loss. These formulations are tested in a synthetic history matching problem with channelized facies. Secondly, we propose two strategies to allow the use of distance-based localization with the deep learning parameterizations.

Reservoir history matching by ensemble smoother with principle component and sensitivity analysis for heterogeneous formations

Ensemble-based data assimilation methods have been widely investigated and applied for inverse problems of fluid flow in porous media during the past decades. Among these methods, the ensemble Kalman filter and the ensemble smoother are probably the most popular in history-matching applications. For large-scale problems, the ensemble size is limited to the computational resources in running the forward simulation, and is usually much smaller than the total number of gridblocks. In this case, the ensemble forms a reduced-order subspace, and principle component analysis is suggested by retaining the leading eigenvectors after truncation.In this paper, we propose to improve the ensemble smoother by proper orthogonal decomposition with optimal parameter selection. The motivation behind this method is that the estimation would be more accurate if the subspace spanned by the ensemble favors the dominant components. Specifically, the principle component analysis is first applied to obtain the eigenvalues and orthogonal eigenvectors. Then, a number of eigenvectors are selected according to importance from sensitivity analysis, rather than the eigenvalues. Finally, the ensemble smoother is implemented using these eigenvectors.The proposed method is illustrated by a simple mathematical example, and tested in a 2D single-phase and a 3D multiphase flow problem. The numerical results show that the traditional ensemble smoother may provide poor results in some ill-posed inverse problems, because some important features are lost in the randomly generated realizations. The principle component analysis alleviates this issue by sorting the eigenvectors based on energy, but it may still produce unsatisfactory results. The combination of sensitivity analysis and principle component analysis provides more accurate matching and prediction than those obtained from the previous two methods. The local sensitivity analysis is preferred since it is much more efficient than the global sensitivity analysis, and its accuracy is validated in the case studies.

Data-space inversion using a recurrent autoencoder for time-series parameterization

Data-space inversion (DSI) and related procedures represent a family of methods applicable for data assimilation in subsurface flow settings. These methods differ from usual model-based techniques in that they provide only posterior predictions for quantities (time series) of interest, not posterior models with calibrated parameters. DSI methods require a large number (O(500–1000)) of flow simulations to first be performed on prior geological realizations. Given observed data, posterior predictions for time series of interest, such as well injection or production rates, can then be generated directly. DSI operates in a Bayesian setting and provides posterior samples of the data vector. In this work, we develop and evaluate a new approach for data parameterization in DSI. Parameterization is useful in DSI as it reduces the number of variables to determine in the inversion, and it maintains the physical character of the data variables. The new parameterization uses a recurrent autoencoder (RAE) for dimension reduction, and a long short-term memory (LSTM) recurrent neural network architecture to represent flow-rate time series. The RAE-based parameterization is combined with an ensemble smoother with multiple data assimilation (ESMDA) for posterior data sample generation. Results are presented for two- and three-phase flows in a 2D channelized system and a 3D multi-Gaussian model. The new DSI RAE procedure, along with several existing DSI treatments, is assessed through detailed comparison to reference rejection sampling (RS) results. The new DSI methodology is shown to consistently outperform existing approaches, in terms of statistical (P10–P90 interval and Mahalanobis distance) agreement with RS results. The method is also shown to accurately capture derived quantities which are computed from variables considered directly in DSI. This requires correlation and covariance between variables to be properly captured, and accuracy in these relationships is demonstrated. The RAE-based parameterization developed here is clearly useful in DSI, and it may also find application in other subsurface flow problems.

Data assimilation of satellite-based terrestrial water storage changes into a hydrology land-surface model

Accurate estimation of snow mass or snow water equivalent (SWE) over space and time is required for global and regional predictions of the effects of climate change. This work investigates whether integration of remotely sensed terrestrial water storage (TWS) information, which is derived from the Gravity Recovery and Climate Experiment (GRACE), can improve SWE and streamflow simulations within a semi-distributed hydrology land surface model. A data assimilation (DA) framework was developed to combine TWS observations with the MESH (Modélisation Environnementale Communautaire – Surface Hydrology) model using an ensemble Kalman smoother (EnKS). The snow-dominated Liard Basin was selected as a case study. The proposed assimilation methodology reduced bias of monthly SWE simulations at the basin scale by 17.5% and improved unbiased root-mean-square difference (ubRMSD) by 23%. At the grid scale, the DA method improved ubRMSD values and correlation coefficients for 85% and 97% of the grid cells, respectively. Effects of GRACE DA on streamflow simulations were evaluated against observations from three river gauges, where it effectively improved the simulation of high flows during snowmelt season from April to June. The influence of GRACE DA on the total flow volume and low flows was found to be variable. In general, the use of GRACE observations in the assimilation framework not only improved the simulation of SWE, but also effectively influenced streamflow simulations.

A comparison between ES-MDA and restart EnKF for the purpose of the simultaneous identification of a contaminant source and hydraulic conductivity

Understanding a contaminant source may help in a better management and risk assessment of a polluted aquifer. However, contaminant source information may not be available when a pollutant is detected in a drinking well. The restart ensemble Kalman filter (restart EnKF, also named r-EnKF) has been demonstrated in synthetic and laboratory experiments as an efficient solution for the identification of a contaminant source. Recently, the ensemble smoother with multiple data assimilation (ES-MDA) has been proposed as an alternative to the r-EnKF as a more efficient solution given that the r-EnKF needs to restart the simulation of the state equation from time zero after each data assimilation step. An analysis, in a synthetic aquifer, of the accuracy of the ES-MDA for the simultaneous identification of a contaminant source and the spatial distribution of hydraulic conductivity by assimilating both piezometric head and concentration observations is carried out using the r-EnKF as a benchmark. The conclusion is that the ES-MDA can outperform the r-EnKF, but the expected speed advantage, associated with the possibility of assimilating all data at once, does not exist. For the ES-MDA to reach the same level of accuracy as the r-EnKF, the number of multiple data assimilations must be large, and final computing time is similar for both approaches. However, the ES-MDA can do much better than the r-EnKF if the number of iterations increases even further, with the consequent increase of computational cost.

ES-MDA applied to estimate skin zone properties from injectivity tests data in multilayer reservoirs

Estimating reservoir properties, such as the skin factor, is an essential role of injectivity tests. Nevertheless, determining individual layer properties in multilayer systems remains a difficult task. Some techniques have been proposed to compute layer permeabilities and skin factors based on the pressure response of reservoirs under single-phase flow. Regardless, determining layer skin zone permeability and radius remains challenging. Ensemble-based techniques are optimization methods known for its easy implementation and good data match when estimating model parameters. Notably, the ensemble smoother with multiple data assimilation (ES-MDA) has proven to be a powerful parameter estimation tool. Thereby, this work attempts to apply the ES-MDA to determine individual layer skin zone properties. To the best of the authors’ knowledge, this is the first time that ES-MDA is applied to this problem. In all tested cases, the resulting histogram contains the reference parameters. We also presented a sensitivity study regarding the performance of ES-MDA when different number of assimilations and ensemble sizes are used.

Identification of non-Gaussian parameters in heterogeneous aquifers by a modified probability conditioning method through hydraulic-head assimilation

Parameter estimation with uncertainty quantification is essential in groundwater modeling to ensure model quality; however, parameter estimation, especially for non-Gaussian distributed parameters in highly heterogeneous aquifers, is still a great challenge. The ensemble smoother with multiple data assimilation (ES-MDA) is one of the most popular and effective ensemble-based data assimilation algorithms. However, it only works for multi-Gaussian fields, since two-point statistics are used to estimate the co-relation between parameters and state variables. The probability conditioning method (PCM) has the capability to integrate nonlinear flow data into facies simulation, but it has an assumption of homogeneity within each facies. Full characterization of facies and estimates of hydraulic conductivity within each facies are equally important. This work firstly modifies the original PCM, introducing a new probability assignment method, to consider within-facies heterogeneities, and then it is further combined with the ES-MDA to estimate non-Gaussian distributed hydraulic parameters in a groundwater model. The proposed method is evaluated using a two-facies case and a three-facies case in groundwater modeling. Both cases demonstrate that the modified PCM is effective for facies delineation, especially to identify high heterogeneities in each facies, as well as non-Gaussian characteristics with good connectivity within certain facies. The results also show that the performances of data reproduction and model prediction are of high accuracy and low uncertainty, which is attributed to the accurate characterization of the non-Gaussian parameters in the heterogeneous aquifers used.

Feature-Oriented Joint Time-Lapse Seismic and Electromagnetic History Matching Using Ensemble Methods

SummaryWe propose a feature-oriented ensemble history-matching workflow with a focus on the integration of time-lapse seismic and electromagnetic (EM) data. The developed workflow consists of two main steps. First, the rock cross-properties, such as water saturation and porosity relating to both seismic and EM attributes, are estimated by joint inversion of seismic and EM data using an iterative ensemble smoother (ES). Second, the remaining model parameters of interest, such as permeability, are calibrated using the updated cross-properties. To assimilate the inverted-saturation information efficiently, we take a feature-oriented integration approach in which front positions are identified from the inverted-saturation field. Related model parameters are then conditioned to the interpreted fronts using the iterative ES with a distance parameterization. The novelty of the proposed approach consists of combining the feature-oriented history matching with ensemble-based geophysical inversion to achieve an efficient joint integration of multiple sources of geophysical data. The performance of the proposed history-matching workflow is examined using a 2D channelized reservoir model and a more-realistic 3D reservoir model with a crosswell configuration for seismic and EM surveys. It is demonstrated that the developed workflow provides a novel and effective way to calibrate reservoir models with multiple sources of geophysical data. The experimental results show a positive synergy effect on the characterization of model variables by jointly assimilating seismic and EM data.

An Efficient Algorithm for Stochastic Ensemble Smoothing

The state of the environment is assessed with a mathematical model and observational data by using a data assimilation procedure. At present the ensemble Kalman filter is one of the most popular data assimilation algorithms. An important component of the data assimilation procedure is assessment not only of the predicted values, but also of parameters that are not described by the model. A single correction procedure from observational data in the ensemble Kalman filter may not provide the required accuracy. In this regard, an ensemble smoothing algorithm in which data from a certain time interval are used to estimate values at a given time is becoming increasingly popular. This paper considers a generalization of a previously proposed algorithm which is a version of the stochastic ensemble Kalman filter. The generalized algorithm is an ensemble smoothing algorithm in which smoothing is performed for the sample mean, and then the ensemble of perturbations is transformed. The transformation matrix proposed in this paper is used to estimate both the predicted value and the parameter. An important advantage of the algorithm is its locality, which makes it possible to estimate the parameter in a given domain. The paper provides a justification of the applicability of this algorithm to ensemble smoothing. Test calculations are performed with a one-dimensional model of transport and diffusion of a passive pollutant. The algorithm is efficient and can be used to assess the state of the environment.

Improvement of sugarcane yield estimation by assimilating UAV-derived plant height observations

Crop yield estimation plays a significant role in agricultural development and management decision making. Simulation of crop growth at field scales requires high-resolution remote sensing (RS) data and an applicable model. However, RS data based on satellite sensors are affected by atmospheric interference, limiting its use in precision agriculture. The objective of this study is to assimilate plant height data into the crop model to improve yield estimation and optimize agricultural water management. The plant height data are monitored regularly in 60 plots at Chongzuo experimental station (Guangxi, China) from an Unmanned Aerial Vehicle (UAV) system and ground-based measurements through a two-year sugarcane experiment (2016 and 2017). A SWAP-PH model (i.e., SWAP model considering plant height simulation) coupled with an iterative ensemble smoother algorithm is constructed to simulate plant height and crop growth procedure. We analyze its performance under the effects of different observation errors, ensemble sizes and development stages on crop growth and yield estimation, and compare the plant height assimilation results from two types of observation platforms. Our results reveal that the crop growth simulation at field scales is improved by incorporating the plant height data into the model. Moreover, incorporating the plant height measurements in the late growth stage with an absolute error of 1–2 cm and ensemble size over 50 could achieve an acceptable crop yield estimation result. In addition, assimilating the UAV-derived measurements could contribute to better yield estimation results than ground-measured plant height data. Finally, we propose a supplemental irrigation and drainage strategy in July and August to maximize the sugarcane yield in Guangxi region.

Prediction of CO₂ Saturation Spatial Distribution Using Geostatistical Inversion of Time-Lapse Geophysical Data

Carbon dioxide sequestration in deep saline aquifers and depleted reservoirs relies on numerical models for the prediction of the spatial distribution of CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> saturation during injection and migration. Due to the limited knowledge of the rock and fluid properties before injection, model predictions are often uncertain and must be updated when new measurements are available. The spatial distribution of CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> saturation and the plume location can be monitored using time-lapse geophysical data, such as seismic and controlled source electromagnetic surveys. We propose a geostatistical inversion approach for the prediction of the time-dependent spatial distribution of CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> saturation from geophysical data. The methodology is based on the application of a stochastic optimization method, the Ensemble Smoother, for the solution of the inverse problem, using rock physics and geophysical models. The inversion is applied to the difference in the geophysical data acquired before and during injection. The predicted models of CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> saturation are obtained by updating an ensemble of geostatistically generated prior realizations, based on the misfit between geophysical model predictions and measured data. The novelty of the approach is the integration of geostatistical algorithms and stochastic optimization methods for the joint inversion of geophysical data. The proposed approach allows including hydrological constraints in the prior model and quantifying the prediction uncertainty due to the noise and resolution of the data and approximations in the physical relations. The method is applied to the Johansen formation model, offshore Norway, using synthetic seismic and electromagnetic data.

Coupling ensemble smoother and deep learning with generative adversarial networks to deal with non-Gaussianity in flow and transport data assimilation

Dynamic data such as hydraulic head and concentration data can be integrated into the groundwater flow and contaminant transport model to improve its predictive ability for groundwater resource management and aquifer remediation. Ensemble Smoother with Multiple Data Assimilation (ES-MDA) has gained popularity for data assimilation in the field of hydrogeology, where aquifer parameters such as hydraulic conductivity are calibrated by conditioning on observed dynamic data. The ES-MDA has an optimal solution if aquifer parameters follow a multi-Gaussian distribution. However, fluvial deposits commonly exhibit a strong heterogeneity with channels (i.e., connectivity). In other words, the hydraulic conductivity does not follow the multi-Gaussian distribution. To deal with data assimilation in channelized aquifers, we propose to couple ES-MDA with deep learning. Specifically, Generative Adversarial Networks (GAN), a deep learning algorithm, are used to re-parameterize the channelized aquifer with a low-dimension latent variable. The ES-MDA is then used to update the latent variable by assimilating dynamic data into the groundwater model. Synthetic studies of groundwater flow and contaminant transport models are used to demonstrate the proposed method. The results illustrate that the coupling of GAN and ES-MDA is able to reconstruct the channel structures and reduce the uncertainty of hydraulic head and contaminant concentration predictions.

4D seismic history matching: Assessing the use of a dictionary learning based sparse representation method

It is possible to improve oil-reservoir simulation models by conditioning them on 4D seismic data. Computational issues may arise related to both storage and CPU time due to the size of the 4D seismic dataset. An approach to reducing the computational requirements is to use a sparse representation method, e.g., Dictionary Learning, to select only the main features of the 4D seismic data. However, the introduction of a sparse representation method incurs an additional computational cost. Alternatively, if one uses ensemble-based methods, it is possible to reduce storage and CPU time by projecting the full seismic dataset on a smaller subspace. This paper evaluates the potential of sparsely representing the seismic data. We compare two experiments, one where we condition on the full dataset projected on a smaller subspace, and one where we use Dictionary Learning to represent the data sparsely. We use Dictionary Learning both on the complete 4D seismic dataset and also on a denoised version of the data. We perform the data assimilation in a slightly different formulation of the Iterative Ensemble Smoother Regularized Levenberg–Marquardt together with correlation-based adaptive localization. We apply these methods to the Brugge benchmark case. Experiment results show that sparse representation methods lead to a final ensemble that is closer to the reference solution, and denoising the seismic data before applying the sparse representation allows us to capture the 4D effect better. Thus, using a sparse representation method in 4D-seismic history matching leads to improved results compared to what we obtain when conditioning the models on the projected 4D seismic dataset.

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.

Development and Calibration of a Semianalytic Model for Shale Wells with Nonuniform Distribution of Induced Fractures Based on ES-MDA Method

Given reliable parameters, a newly developed semianalytic model could offer an efficient option to predict the performance of the multi-fractured horizontal wells (MFHWs) in unconventional gas reservoirs. However, two major challenges come from the accurate description and significant parameters uncertainty of stimulated reservoir volume (SRV). The objective of this work is to develop and calibrate a semianalytic model using the ensemble smoother with multiple data assimilation (ES-MDA) method for the uncertainty reduction in the description and forecasting of MFHWs with nonuniform distribution of induced fractures. The fractal dimensions of induced-fracture spacing (dfs) and aperture (dfa) and tortuosity index of induced-fracture system (θ) are included based on fractal theory to describe the properties of SRV region. Additionally, for shale gas reservoirs, gas transport mechanisms, e.g., viscous flow with slippage, Knudsen diffusion, and surface diffusion, among multi-media including porous kerogen, inorganic matter, and fracture system are taken into account and the model is verified. Then, the effects of the fractal dimensions and tortuosity index of induced fractures on MFHWs performances are analyzed. What follows is employing the ES-MDA method with the presented model to reduce uncertainty in the forecasting of gas production rate for MFHWs in unconventional gas reservoirs using a synthetic case for the tight gas reservoir and a real field case for the shale gas reservoir. The results show that when the fractal dimensions of induced-fracture spacing and aperture is smaller than 2.0 or the tortuosity index of induced-fracture system is larger than 0, the permeability of induced-fracture system decreases with the increase of the distance from hydraulic fractures (HFs) in SRV region. The large dfs or small θ causes the small average permeability of the induced-fracture system, which results in large dimensionless pseudo-pressure and small dimensionless production rate. The matching results indicate that the proposed method could enrich the application of the semianalytic model in the practical field.

Integrated hydrogeophysical modelling and data assimilation for geoelectrical leak detection

Time-lapse electrical resistivity tomography (ERT) measurements provide indirectobservations of hydrological processes in the Earth's shallow subsurface at high spatial and temporal resolution. ERT has been used in the past decades to detect leaks and monitor the evolution of associated contaminant plumes. Specifically, inverted resistivity images allow visualization of the dynamic changes in the structure of the plume. However, existing methods do not allow the direct estimation of leak parameters (e.g. leak rate, location, etc.) and their uncertainties. We propose an ensemble-based data assimilation framework that evaluates proposed hydrological models against observed time-lapse ERT measurements without directly inverting for the resistivities. Each proposed hydrological model is run through the parallel coupled hydro-geophysical simulation code PFLOTRAN-E4D to obtain simulated ERT measurements. The ensemble of model proposals is then updated using an iterative ensemble smoother. We demonstrate the proposed framework on synthetic and field ERT data from controlled tracer injection experiments. Our results show that the approach allows joint identification of contaminant source location, initial release time, and solute loading from the cross-borehole time-lapse ERT data, alongside with an assessment of uncertainties in these estimates. We demonstrate a reduction in site-wide uncertainty by comparing the prior and posterior plume mass discharges at a selected image plane. This framework is particularly attractive to sites that have previously undergone extensive geological investigation (e.g., nuclear sites). It is well suited to complement ERT imaging and we discuss practical issues in its application to field problems.

Stochastic inversion method of time-lapse controlled source electromagnetic data for CO2 plume monitoring

Carbon dioxide storage in deep saline aquifers can potentially reduce the CO2 concentration in the atmosphere due to anthropogenic activities with limited environmental impact. To minimize the risks of leakage, it is necessary to monitor the CO2 distribution in the reservoir. Here, we present a stochastic optimization method namely the Ensemble Smoother, to invert time-lapse marine controlled source electromagnetic data for predicting the CO2 plume location. Electromagnetic surveys have been successfully used in subsurface monitoring studies in different geosciences applications due to their sensitivity to the changes in the fluid in porous rocks. The inverse method is based on a Bayesian approach, in which an ensemble of stochastically generated prior models representing the spatial distribution of CO2 saturation is updated according to the mismatch between the measured data and the predicted electromagnetic response of the prior models. The proposed method generates an ensemble of updated realizations, the posterior models of the spatial distribution of CO2 saturation, that match the observations. The variance of the posterior models represents the uncertainty of the CO2 distribution. Compared to deterministic inversion methods, our proposed method is better suited to solve non-linear inverse problems with the added benefit of uncertainty quantification. The method is tested and validated on a real dataset, the Johansen formation, offshore Norway, for which we created synthetic time-lapse electromagnetic data for the entire injection period. The inversion of electromagnetic data shows that the proposed method can accurately predict the prediction of the CO2 plume location and quantify the associated uncertainty.