Ensemble Smoother With Multiple Data Assimilation Research Articles

Joint identification of groundwater pollution source information, model parameters, and boundary conditions based on a novel ES-MDA with a wheel battle strategy

Groundwater pollution source identification (GPSI) is an important prerequisite for pollution remediation and risk assessment. However, an accurate GPSI is usually difficult to achieve due to the need to simultaneously consider practical problems and theoretical research. For practical problems, the boundary conditions, especially the concentration boundary, are often given as known constants through prior information. However, in most practical situations, the boundary conditions are complex and cannot be accurately estimated in advance, which leads to the distortion of the final identification results. Therefore, this study focused on the concentration boundary, and first proposed to jointly identify three types of unknown variables (source information, model parameters, and boundary conditions) to ensure that the identification results had more practical value. For theoretical research, as the number of unknown variable types increases, the difficulty of solving the inverse problem often increases, which may lead to inaccurate inversion results. Thus, a novel ensemble smoother with multiple data assimilation (ES-MDA) with a wheel battle strategy was proposed, enhancing the identification accuracy. We designed two cases to verify the effectiveness and practicality of the above ideas: a low-dimensional Case 1 (including four different synthetic scenarios: three scenarios with unknown boundary conditions under three concentration boundary modes and a scenario with known boundary conditions) and a high-dimensional complex Case 2. Identifying the boundary conditions in GPSI was found to be of great significance. Compared to the standard ES-MDA method, the ES-MDA with a wheel battle strategy proposed here could improve the inversion accuracy, and had certain effectiveness and practicality.

History matching for geological carbon storage using data-space inversion with spatio-temporal data parameterization

History matching based on monitoring data will enable uncertainty reduction, and thus improved aquifer management, in industrial-scale carbon storage operations. In traditional model-based data assimilation, geomodel parameters are modified to force agreement between flow simulation results and observations. In data-space inversion (DSI), history-matched quantities of interest, e.g., posterior pressure and saturation fields conditioned to observations, are inferred directly, without constructing posterior geomodels. This is accomplished efficiently using a set of O(1000) prior simulation results, data parameterization, and posterior sampling within a Bayesian setting. In this study, we develop and implement (in DSI) a deep-learning-based parameterization to represent spatio-temporal pressure and CO2 saturation fields at a set of time steps. The new parameterization uses an adversarial autoencoder (AAE) for dimension reduction and a convolutional long short-term memory (convLSTM) network to represent the spatial distribution and temporal evolution of the pressure and saturation fields. This parameterization is used with an ensemble smoother with multiple data assimilation (ESMDA) in the DSI framework to enable posterior predictions. A realistic 3D system characterized by prior geological realizations drawn from a range of geological scenarios is considered. A local grid refinement procedure is introduced to estimate the error covariance term that appears in the history matching formulation. Extensive history matching results are presented for various quantities, for multiple synthetic true models, using the new DSI framework. Substantial uncertainty reduction in posterior pressure and saturation fields is achieved in all cases. The framework is also applied to efficiently provide posterior predictions for a range of error covariance specifications. Such an assessment would be very expensive using a traditional model-based approach.

A Comparison of Inversion Methods for Surrogate‐Based Groundwater Contamination Source Identification With Varying Degrees of Model Complexity

AbstractAccurate identification of groundwater contamination sources is important for designing efficacious site remediation strategies. Currently, the methods for identifying contamination sources mainly fall into three distinct categories: simulation optimization, Bayesian inference, and data assimilation. Each method has its own advantages and disadvantages under specific site conditions. To evaluate the applicability of these methods, we chose one representative inversion algorithm from each category, namely the Improved Butterfly Optimization Algorithm (IBOA) for simulation optimization, the Ensemble Smoother with Multiple Data Assimilation (ES‐MDA) for data assimilation, and the DiffeRential Evolution Adaptive Metropolis with a Snooker Update and Sampling from a Past Archive (DREAM(ZS)) for Bayesian inference. We conducted a comprehensive evaluation of these methods' performance under different model complexities, employing a surrogate model as a substitute for the complex forward model. By addressing two distinct problems involving conservative pollutant transport and Light Non‐Aqueous Phase Liquid (LNAPL) transport with biodegradation, we employed four criteria (elapsed time, result accuracy, posterior probability distribution, and noise resistance) for evaluation. The findings unequivocally indicate that DREAM(ZS) outperforms others in terms of result accuracy and posterior probability distribution. It also adeptly navigates the interrelations among disparate unknown variables. The strength of ES‐MDA lies in its efficiency. It achieves relatively satisfactory results with a reduced computational burden. In contrast, IBOA underperforms in both test problems. In terms of resistance to noise, both DREAM(ZS) and ES‐MDA perform better than IBOA does.

Leveraging deep learning with progressive growing GAN and ensemble smoother with multiple data assimilation for inverse modeling

The incorporation of hard data in geostatistical modeling is crucial for enhancing the accuracy of interpolating or stochastically estimating subsurface spatial features. The hard data at specified points in the model domain serve as a guide in optimizing the unknown parameters to follow the patterns of the hard data. Recently, a novel approach to solving hydrogeologic/reservoir modeling problems has emerged by using deep generative models, specifically generative adversarial networks (GANs), to generate realistic and diverse images of channelized aquifers. This subsequently can be coupled with inverse models to solve parameter estimation problems. This study focused on using an improved GAN, called a progressive growing generative adversarial network (PGGAN), conditioned with hard data to perform parameter estimation of complex facies models by coupling an ensemble smoother with multiple data assimilation (ES-MDA). First, the PGGAN was trained to an image with 128 × 128 resolution. The trained PGGAN was used to generate hydraulic conductivity fields when fed an ensemble of latent variables and hard data. The ES-MDA then was used to update the latent variable with the help of hydraulic head data obtained from the groundwater model. The approach was tested on synthetic hydraulic conductivity data. Results show that this approach was able to perform efficient estimation of an unknown facies model domain. Additionally, the proposed method was applied to a different test case of a facies model exhibiting different statistical characteristics. The results were satisfactory, demonstrating that the method is not constrained to the particular hydraulic conductivity fields introduced in the generator’s training.

Comparison of three recent discrete stochastic inversion methods and influence of the prior choice

Groundwater flow depends on subsurface heterogeneity, which often calls for categorical fields to represent different geological facies. The knowledge about subsurface is however limited and often provided indirectly by state variables, such as hydraulic heads of contaminant concentrations. In such cases, solving a categorical inverse problem is an important step in subsurface modeling. In this work, we present and compare three recent inverse frameworks: Posterior Population Expansion (PoPEx), Ensemble Smoother with Multiple Data Assimilation (ESMDA), and DREAM-ZS (a Markov chain Monte Carlo sampler). PoPEx and ESDMA are used with Multiple-point statistics (MPS) as geostatistical engines, and DREAM-ZS is used with a Wasserstein generative adversarial network (WGAN). The three inversion methods are tested on a synthetic example of a pumping test in a fluvial channelized aquifer. Moreover, the inverse problem is solved three times with each method, each time using a different training image to check the performance of the methods with different geological priors. To assess the quality of the results, we propose a framework based on continuous ranked probability score (CRPS), which compares single true values with predictive distributions. All methods performed well when using the training image used to create the reference, but their performances were degraded with the alternative training images. PoPEx produced the least geological artifacts but presented a rather slow convergence. ESMDA showed initially a very fast convergence which reaches a plateau, contrary to the remaining methods. DREAM-ZS was overly confident in placing some incorrect geological features but outperformed the other methods in terms of convergence.

Feature Extraction in Time-Lapse Seismic Using Deep Learning for Data Assimilation

Summary Assimilation of time-lapse (4D) seismic data with ensemble-based methods is challenging because of the massive number of data points. This situation requires excessive computational time and memory usage during the model updating step. We addressed this problem using a deep convolutional autoencoder to extract the relevant features of 4D images and generate a reduced representation of the data. The architecture of the autoencoder is based on the VGG-19 network, a deep convolutional architecture with 19 layers well-known for its effectiveness in image classification and object recognition. Some advantages of VGG-19 are the possibility of using some pretrained convolutional layers to create a feature extractor and taking advantage of the transfer learning technique to address other related problem domains. Using a pretrained model bypasses the need for large training data sets and avoids the high computational demand to train a deep network. For further improvements in the reconstruction of the seismic images, we apply a fine-tuning of the weights of the latent convolutional layer. We propose to use a fully convolutional architecture, which allows the application of distance-based localization during data assimilation with the ensemble smoother with multiple data assimilation (ES-MDA). The performance of the proposed method is investigated in a synthetic benchmark problem with realistic settings. We evaluate the methodology with three variants of the autoencoder, each one with a different level of data reduction. The experiments indicate that it is possible to use latent representations with major data reductions without impairing the quality of the data assimilation. Additionally, we compare central processing unit (CPU) and graphics processing unit (GPU) implementations of the ES-MDA update step and show in another synthetic problem that the reduction in the number of data points obtained with the application of the deep autoencoder may provide a substantial improvement in the overall computation cost of the data assimilation for large reservoir models.

Seismic fluid identification method based on the joint PP‐ and SH–SH‐wave stochastic inversion

AbstractPre‐stack seismic inversion is an effective method for elastic parameter inversion using seismic data, which facilitates seismic fluid identification. However, pure PP‐wave inversion has issues of strong multi‐solution and limited prediction accuracy. Therefore, we propose a seismic fluid identification approach based on the joint PP‐ and SH–SH‐wave stochastic inversion. First, the linearized SH–SH‐wave amplitude variation with offset approximation parameterized by shear modulus and density is derived. Numerical simulations demonstrate that the SH–SH‐wave amplitude variation with offset approximation has a good accuracy. Reflection coefficient contribution analysis indicates that the new formulation has better parameter sensitivity to shear modulus and density than the PP wave amplitude variation with offset approximation derived by Russell, which helps one to improve the inversion of shear modulus and density. On this basis, we construct a joint inversion equation of PP and SH–SH waves for a Russell fluid indicator, a shear modulus and density and present a novel joint stochastic inversion method based on the ensemble smoother with multiple data assimilation. Stanford VI‐E model tests reveal that the Russell fluid indicator factor, shear modulus and density obtained from the joint PP‐ and SH–SH‐wave inversion have higher identification accuracy and smaller relative errors than those from pure PP‐wave inversion. Furthermore, field data tests indicate that this method has practical applicability in seismic fluid identification.

Contaminant source and aquifer characterization: An application of ES-MDA demonstrating the assimilation of geophysical data

Contaminant source and aquifer characterization (CSAC) is critical in groundwater pollution evaluation and remediation. The ensemble smoother with multiple data assimilation (ES-MDA) is utilized to jointly identify contaminant source information and hydraulic conductivities by assimilating time-lapse electrical resistivity tomography (ERT) data. In a synthetic profile with a time-varying release history in a heterogeneous aquifer, we verify the performance of the proposed data assimilation framework. The results show that the CSAC problem could be handled by the proposed approach. The time-varying release history and the high permeability area can be identified with adequate time-lapse ERT measurements. Further reproduction of the evolution of the plume after CSAC also shows consistency with the reference plume. The poor conditioning inversion caused by the filter inbreeding is analyzed by comparing four scenarios with different apparent resistivity measurements. Furthermore, we also evaluate the impact of uncertainties in the petrophysical properties and geophysical observations on our data assimilation framework. The results show that the proposed ES-MDA data assimilation framework could provide a convincing inversion of the time-varying release history and hydraulic conductivities.

Geomechanical Response in Energy Transition Applications: Assessing its Role with Data Assimilation

The subsurface is characterized by significant uncertainties that pose challenges for geothermal energy production, CO2 sequestration, and hydrocarbon field development. To handle this uncertainty, engineers often use ensembles of models with different porosity and permeability distributions. In addition, history-matching procedures can further reduce uncertainty. However, traditional methods using well bottom hole pressures as observations can be challenging for geothermal or CO2 sequestration projects with a limited number of wells. To overcome this limitation, additional observation data can be used. In this study, we propose using vertical displacements to evaluate the subsidence/uplift of the model. Pressure depletion, for example, leads to surface subsidence due to rock compaction. We suggest measuring subsidence during field development and using it as an objective function to minimize the difference from measured data. Our approach provides a more reliable and accurate way to evaluate the model's subsurface behavior, helping to reduce uncertainties in geothermal energy production, CO2 sequestration, or hydrocarbon field development.
 We used a state-of-the-art simulator called DARTS [1] to simulate fluid flow in the subsurface, and computed geomechanics response using a physics-based proxy. We then performed history matching using ensemble smoother with multiple data assimilation (ES-MDA) of the reservoir subsidence to better understand the reservoir properties, such as permeability [2]. Our workflow involved generating an ensemble of models with varying permeability and computing vertical displacements using the physics-based proxy model for geomechanics [3]. We then applied ES-MDA to find the model with permeability that fits best to the reference subsidence surface. Flow pattern in heterogeneous reservoir determines in-situ stress changes, which could induce seismicity in a presence of natural faults. We used fault slippage criteria for each scenario. Data assimilation procedure provide us history matched model, and, therefore, more reliable assessment of induced seismicity risk. We used 2.5D extruded mesh with heterogeneous permeability and uniform geomechanical properties.

On the use of different data assimilation schemes in a fully coupled hydro-mechanical slope stability analysis

ABSTRACT Different data assimilation schemes such as the ensemble Kalman filter (EnKF), ensemble smoother (ES) and ensemble smoother with multiple data assimilation (ESMDA) are implemented in a hydro-mechanical slope stability analysis. For a synthetic case, these schemes assimilate displacements at the crest and the slope to estimate strength and stiffness parameters. These estimated parameters are then used to estimate the system's state and factor of safety (FoS). The results show that EnKF provides an FoS estimation with a mean close to the truth and with the smallest standard deviation, with ESMDA using the largest amount of assimilation steps also providing a mean close to the truth but with less confidence. The ES and ESMDA with fewer assimilation steps underestimate the FoS approximation and have low confidence. Assimilating measurements over a longer period provides a more accurate parameter, state and FoS estimation. ES has the best computational performance, with ESMDA performing worse, with its performance dependent on the number of assimilation steps. The computational performance of the EnKF is better than ESMDA but around 50% worse than the ES. Non-linearity of the underlying problem is a key cause of the multi-step assimilation processes having a better performance.

Frequency-domain electromagnetic induction for the prediction of electrical conductivity and magnetic susceptibility using geostatistical inversion and randomized tensor decomposition

High-resolution characterization of near-surface systems is crucial for a variety of subsurface applications. Frequency-domain electromagnetic (FDEM) induction has been widely used in near-surface characterization compared with other geophysical methods due to its flexibility in acquisition and its ability to survey large areas with high resolution but with relatively low costs. FDEM measurements are sensitive to subsurface electrical conductivity (EC) and magnetic susceptibility (MS). However, the prediction of these properties requires solving a geophysical inverse problem. We combine ensemble smoother with multiple data assimilation (ES-MDA) and model reparameterization via randomized tensor decomposition (RTD) to simultaneously predict EC and MS from measured FDEM data. ES-MDA is an iterative data assimilation method that can be applied to nonlinear forward operators and provides multiple posterior realizations conditioned on the geophysical measurements to evaluate the model uncertainty. However, its application is usually computationally prohibitive for large-scale 3D problems. To overcome this limitation, we reduce the model parameters using RTD and then perform the inversion in the low-dimensional model space. The method is applied to synthetic and noisy real data sets. In the synthetic application example, the predicted posterior realizations illustrate the ability of our method to recover the true models of EC and MS accurately. The real case application comprises FDEM data acquired over arable land characterized by quaternary siliciclastic deposits with geoarchaeological features. We assess the performance of the inversion method at a borehole location not used to constrain the inversion. The inverted models do capture the available log data, illustrating the applicability of the inversion method to noisy real data.

Experimental sandbox tracer tests to characterize a two-facies aquifer via an ensemble smoother

Estimating aquifer properties and their spatial variability is the most challenging part of groundwater flow and transport simulations. In this work, an ensemble Kalman-based method, the ensemble smoother with multiple data assimilation (ES-MDA), is applied to infer the characteristics of a binary field by means of tracer test data collected in an experimental sandbox. Two different approaches are compared: the first one aims at estimating the hydraulic conductivity over the whole field assuming that the rest of the hydraulic and transport parameters are known by applying the standard ES-MDA method; the second one couples the ES-MDA with a truncated Gaussian model to simultaneously estimate the spatial distribution of two geological lithotypes and their main hydraulic and transport properties. Both procedures are tested following a fully parameterized approach and a pilot point approach. A synthetic case that mimics the sandbox experiment was developed to test the capability of the proposed methods and find out their optimal configurations to be used for the real case. The results show that the ES-MDA coupled with a truncated Gaussian model outperforms the standard ES-MDA and it reproduces well the binary field and the aquifer properties also in the presence of large measurement errors. The fully parametrized and pilot point approaches lead to comparable solutions, with less computation time required by the pilot point approach.

Analytical solutions of fractal and anomalous diffusion models for pressure and rate transient analysis

In this study, we examine the modeling of the naturally fractured reservoirs based on the fractal, Metzler, and Raghavan anomalous diffusion models which are different from Darcy flux law. The governing equations of these models are presented for a vertical well producing at either a constant rate (CR) or constant bottomhole pressure (CBHP) production in an infinite-acting reservoir. New analytical and early- and late-time asymptotic approximate solutions were derived for the finite-wellbore problem for the Raghavan anomalous model. The approximate solutions identify the fractal and anomalous diffusion parameters that influence behaviors of the early- and late-time pressure and rate transient data. The method of Laplace transformation is used to find the solutions in dimensionless forms which are then inverted to the real-time domain using a numerical inversion algorithm. In addition, we compare the solutions for the fractal, Metzler anomalous, and Raghavan anomalous models and delineate the similarities and differences among these solutions. We show the effect of fractal parameters on diffusion models, the relationship between the fractal model, and Raghavan and Metzler anomalous models, in terms of dimensionless pressure and dimensionless pressure derivative responses in the reservoir. We show that the “dimensionless” pressure defined by Raghavan is not a dimensionless quantity unless the Euclidean dimension d=2 (or for a fractal structure with the mass fractal dimension df=2). To the best of our knowledge, in the literature, there exists no study providing a detailed review and comparing the applicability of each diffusion model on real field well-test data as well as providing new analytical solutions and early- and late-time asymptotic approximate solutions for the anomalous diffusion for both the CR and CBHP cases for the finite-wellbore problem. Hence, this study fills this gap. In addition, we apply the three diffusion models to an interference well test data from the Kizildere geothermal reservoir in Turkey by using the ensemble smoother with multiple data assimilation (ES-MDA) method. The results show that the fractal model in terms of the goodness of fit (root-mean-square error – RMSE) best matches the field test data.

Data Assimilation of Production and Multiple 4D Seismic Acquisitions in a Deepwater Field Using Ensemble Smoother with Multiple Data Assimilation

Summary In recent years, time-lapse (4D) seismic (4DS) data have been widely used for reservoir monitoring to provide relevant information on dynamic changes occurring during production. In complex reservoirs, multiple seismic monitor surveys are usually available. Updating reservoir models with these data can be very beneficial to improve the field’s management. In the quantitative integration of 4DS data into the data assimilation (DA) process, it is crucial to define how to deal with more than one seismic monitor. In this work, we continue a series of investigations about seismic DA procedures and expand on them by analyzing ways to assimilate more than one seismic monitor. More specifically, we evaluate different ways of using production data and two monitor surveys (M3 and M5) to calibrate the dynamic models of a real Brazilian reservoir using the ensemble smoother with multiple data assimilation (ES-MDA) method. We performed the following experiments: (1) sequential assimilation of M3 and M5 with parts of well history divided according to the seismic acquisition dates; (2) assimilation of M3 with the entire well history and subsequent assimilation of M5; (3) assimilation of well and M3 data; and (4) assimilation of well and M5 data. For comparison purposes, we also assimilated only well data. From the results, we observed that well and 4DS data misfits were reduced when assimilating both monitors, compared to the cases where only a single monitor (any of them) was used with production data. This conclusion is also true in the comparison with results obtained when only assimilating well data. This indicates that both seismic monitors are important data to be quantitatively considered in DA. In this particular field, using a previous DA run to solely assimilate the newly available monitor (Case 2) delivered better models and long-term forecasts. Therefore, this would be our recommendation. This study highlights the importance of several 4DS acquisitions for reservoir monitoring and management and shows the challenges of their application in seismic DA for better life cycle field applications.

Identification of the inflow source in a foul sewer system through techniques of inverse modelling

Infiltration and illegal inflow into foul sewer systems can cause different problems such as a decrease in the performance of treatment plants, the surcharge of pipelines and more frequent overflows, which cause negative impacts on the environment. Water companies are increasingly been driven to address these problems by reducing infiltrations and identifying the sources of illegal inflows. Overall, the traditional techniques applied in these cases are expensive and time consuming and many times only partially efficient. Examples are the use of CCTV inspections, smoke tests and the installation of a large set of sensors to collect continuous data such as flow rates, water levels, temperature or concentrations of pollutants. The aim of this study is to apply two types of inverse numerical techniques to identify the source location of illegal inflows into wastewater systems based on information collected at the outlet of the drained basin and a calibrated numerical model of the sewer network. In this work, the numerical model is developed using the Storm Water Management Model (SWMM) software distributed by the Environmental Protection Agency (USA). We considered a realistic foul sewer system with known dimensional and hydrological characteristics. Synthetic case studies are set up to test the inverse approaches. Assuming a hypothetical rainfall event and an illegal inflow released at a certain location in the sewer system, the numerical model is run forward to obtain the flow hydrograph at the network outlet. This information is then used as available observations to perform the inverse modelling. The first investigated technique is an artificial neural network (ANN) of the feed-forward type. It will be trained to recover the inflow source using the simulation results of SWMM driven by a large set of rainfall events and inflows located at different positions in the sewer network. Once trained, the ANN will be used to identify the location of the inflow based the observed flood wave. The second procedure derives from Kalman filter techniques: the Ensemble Smoother with Multiple Data Assimilation (ES-MDA). Also in this case, the method, starting from the known rainfall event and the observed flow hydrograph, is used to locate the inflow source. In addition to the results of the synthetic case obtained by means of the two procedures, the field applicability to real case studies will be discussed.

Reservoir Modeling, History Matching, and Characterization with a Reservoir Graph Network Model

Summary Efficient reservoir models are more desirable for fast-paced reservoir management. Moreover, due to the complexity of flow underground, it is also essential to capture the fundamental physics for model reliability. Although they are fast, pure data-driven models frequently have issues associated with interpretability, physical consistency, and ability to forecast. On the other hand, we have used full-physics simulation models to mimic and investigate hydrocarbon systems for over several decades. However, considering its infrequent model updates related to high model complexity, it is a big challenge to manage reservoirs using full-physics models in short cycles. The objective here is to propose an approach that blends reservoir physics with data-driven models to fit in the framework of dynamic reservoir management. We propose to use a reservoir graph network (RGNet) modeling approach based on a diffusive time-of-flight (DTOF) concept to simulate reservoir behaviors. By assimilating field observation data (such as pressure and rates), an RGNet model can be used for future predictions, scenario studies, and well-control optimizations. By discretizing DTOF of a 3D system with multiple wells, RGNet simplifies the system into a graph network represented by a set of 1D grid blocks that significantly reduces the system complexity and run time. RGNet can also handle multiple flow problems with various types of physics. In this work, we propose to use two methods to develop reliable and parsimonious models scalable to large-scale systems. In addition, we propose a more robust method to assimilate pressure data. We applied the proposed approach to a synthetic and a field example. Two different history-matching algorithms, the ensemble smoother with multiple data assimilation (ES-MDA) and an adjoint-based method, are compared. While ES-MDA provides the capability for uncertainty analysis, an adjoint-based method generally requires fewer simulation runs to generate a posterior model. With the proposed methods for generating interwell connections, RGNet model calibration can be achieved without system redundancy and spurious long-distance well connectivity. Also, by using a more stable pressure-matching technique, we show that pressure data are better matched and reservoir volume is accurately characterized. RGNet provides a novel hybrid physics and data-driven reservoir modeling method to fit in closed-loop reservoir management (CLRM). As RGNet models are combined with fundamental flowing physics, the calibrated model parameters are easy to interpret and understand. An RGNet model runs with far less computational cost than required by a full-physics model, which allows it to be a more practical solution to history match, predict, and optimize real assets.

Reconstructing the release history of a contaminant source with different precision via the ensemble smoother with multiple data assimilation

Identifying a contaminant time-varying release history is an ill-posed problem but crucial for groundwater contamination issues. A precise inversed release history offers a promising estimation of contaminant movement and is of great importance for environmental monitoring and further management. In this paper, a recent emerging data assimilation method, the ensemble smoother with multiple data assimilation (ES-MDA) is employed to handle this conundrum. The study starts with some synthetic cases in which several factors are analyzed, such as the observation data frequency, covariance inflation schemes, iteration numbers used in the ES-MDA for the purpose of identifying a time-varying contaminant injection event with different precision. The results show that the ES-MDA performs well in recovering the release history when the injection is discretized into 50 or 100-time steps but encounters fluctuation problems in the cases with 300-time steps. Further comparison reveals that the observation data frequency is a very influential factor, while the number of iterations or the kind of covariance inflation used has a lesser effect. Nevertheless, this is a first test in a non-synthetic environment, in which the ES-MDA has proven its ability to recover the release history in two close-to-reality sandbox experiments. The outcome shows that the ES-MDA with Rafiee’s inflation scheme has the ability to capture the main pattern of the release history. But in order to move one more step to field cases, a more detailed description of uncertainties or elaborated parameterization of the time functions is paramount.

Identification of hydraulic conductivity field of a karst aquifer by using transition probability geostatistics and discrete cosine transform with an ensemble method

AbstractKnowledge of the spatial distribution characteristics of hydraulic parameters is essential for the management and protection of karst groundwater resources. In this study, we propose a workflow integrating the transition probability geostatistics (T‐PROGS) and the discrete cosine transform (DCT) with the ensemble smoother with multiple data assimilation (ES‐MDA) method to map the hydraulic conductivity of karst aquifers that usually follow a multimodal distribution. The priori parameter ensemble is constructed from the T‐PROGS and transformed into an approximately Gaussian distributed coefficient field containing critical spatial structural features about the aquifer using DCT. The ES‐MDA method is employed to update the DCT coefficients by assimilating the measurements. In addition, a postprocessing process based on cumulative distribution function (CDF) mapping is used to address the problem of parameters gradually tending to a Gaussian distribution after updating using the ES‐MDA and the inappropriate selection of initial parameter values. In practice, the limited amount of available data makes it difficult to fully capture the spatial distribution of the parameters in the initial ensemble of a single realization. Therefore, we suggest a new strategy for structuring the initial ensemble by mixing samples from multiple realizations. We then apply the proposed approach to four single realization models and a combined multiple realizations model in a field hydraulic tomography investigation of a real karst aquifer. The computed results show that this method can effectively identify the characteristics of the spatial distribution of hydraulic conductivity in karst aquifers. Compared with an individual realization model, the uncertainty of the hydraulic conductivity estimated by the combined multiple realizations model is significantly reduced. Therefore, including more uncertainty of the aquifer in the initial ensemble is beneficial to improve the accuracy of the parameter estimation.

Stochastic multi-fidelity joint hydrogeophysical inversion of consistent geological models

In Quaternary deposits, the characterization of subsurface heterogeneity and its associated uncertainty is critical when dealing with groundwater resource management. The combination of different data types through joint inversion has proven to be an effective way to reduce final model uncertainty. Moreover, it allows the final model to be in agreement with a wider spectrum of data available on site. However, integrating them stochastically through an inversion is very time-consuming and resource expensive, due to the important number of physical simulations needed. The use of multi-fidelity models, by combining low-fidelity inexpensive and less accurate models with high-fidelity expensive and accurate models, allows one to reduce the time needed for inversion to converge. This multiscale logic can be applied for the generation of Quaternary models. Most Quaternary sedimentological models can be considered as geological units (large scale), populated with facies (medium scale), and finally completed by physical parameters (small scale). In this paper, both approaches are combined. A simple and fast time-domain EM 1D geophysical direct problem is used to first constrain a simplified stochastic geologically consistent model, where each stratigraphic unit is considered homogeneous in terms of facies and parameters. The ensemble smoother with multiple data assimilation (ES-MDA) algorithm allows generating an ensemble of plausible subsurface realizations. Fast identification of the large-scale structures is the main point of this step. Once plausible unit models are generated, high-fidelity transient groundwater flow models are incorporated. The low-fidelity models are populated stochastically with heterogeneous facies and their associated parameter distribution. ES-MDA is also used for this task by directly inferring the property values (hydraulic conductivity and resistivity) from the generated model. To preserve consistency, geophysical and hydrogeological data are inverted jointly. This workflow ensures that the models are geologically consistent and are therefore less subject to artifacts due to localized poor-quality data. It is able to robustly estimate the associated uncertainty with the final model. Finally, due to the simplification of both the direct problem and the geology during the low-fidelity part of the inversion, it greatly reduces the time required to converge to an ensemble of complex models while preserving consistency.

Deep Learning for Simultaneous Inference of Hydraulic and Transport Properties

AbstractIdentification of a heterogeneous conductivity field and reconstruction of a contaminant release history are key aspects of subsurface remediation. These two goals are achieved by combining model predictions with sparse and noisy hydraulic head and concentration measurements. Solution of this inverse problem is notoriously difficult due to, in part, high dimensionality of the parameter space and high computational cost of repeated forward solves. We use a convolutional adversarial autoencoder (CAAE) to parameterize a heterogeneous non‐Gaussian conductivity field via a low‐dimensional latent representation. A three‐dimensional dense convolutional encoder‐decoder (DenseED) network serves as a forward surrogate of the flow and transport model. The CAAE‐DenseED surrogate is fed into the ensemble smoother with multiple data assimilation (ESMDA) algorithm to sample from the Bayesian posterior distribution of the unknown parameters, forming a CAAE‐DenseED‐ESMDA inversion framework. The resulting CAAE‐DenseED‐ESMDA inversion strategy is used to identify a three‐dimensional contaminant source and conductivity field. A comparison of the inversion results from CAAE‐ESMDA with physical flow and transport simulator and from CAAE‐DenseED‐ESMDA shows that the latter yields accurate reconstruction results at the fraction of the computational cost of the former.