Towards a hierarchical parametrization to address prior uncertainty in ensemble-based data assimilation

Ensemble-based methods are becoming popular assisted history matching techniques with a growing number of field applications. These methods use an ensemble of model realizations, typically constructed by means of geostatistics, to represent the prior uncertainty. The performance of the history matching is very dependent on the quality of the initial ensemble. However, there is a significant level of uncertainty in the parameters used to define the geostatistical model. From a Bayesian viewpoint, the uncertainty in the geostatistical modeling can be represented by a hyper-prior in a hierarchical formulation. This paper presents the first steps towards a general parametrization to address the problem of uncertainty in the prior modeling. The proposed parametrization is inspired in Gaussian mixtures, where the uncertainty in the prior mean and prior covariance is accounted by defining weights for combining multiple Gaussian ensembles, which are estimated during the data assimilation. The parametrization was successfully tested in a simple reservoir problem where the orientation of the major anisotropic direction of the permeability field was unknown.     

Robust optimisation of CO2 sequestration strategies under geological uncertainty using adaptive sparse grid surrogates

Geologic CO₂ sequestration in deep saline aquifers is a promising technique to mitigate the effect of greenhouse gas emissions. Designing optimal CO₂ injection strategy becomes a challenging problem in the presence of geological uncertainty. We propose a surrogate assisted optimisation technique for robust optimisation of CO₂ injection strategies. The surrogate is built using Adaptive Sparse Grid Interpolation (ASGI) to accelerate the optimisation of CO₂ injection rates. The surrogate model is adaptively built with different numbers of evaluation points (simulation runs) in different dimensions to allow automatic refinement in the dimension where added resolution is needed. This technique is referred to as dimensional adaptivity and provides a good balance between the accuracy of the surrogate model and the number of simulation runs to save computational costs. For a robust design, we propose a utility function which comprises the statistical moment of the objective function. Numerical testing of the proposed approach applied to benchmark functions and reservoir models shows the efficiency of the method for the robust optimisation of CO₂ injection strategies under geological uncertainty.     

Bayesian hierarchical models for soil CO<Subscript>2</Subscript> flux and leak detection at geologic sequestration sites

Proper characterizations of background soil CO₂ respiration rates are critical for interpreting CO₂ leakage monitoring results at geologic sequestration sites. In this paper, a method is developed for determining temperature-dependent critical values of soil CO₂ flux for preliminary leak detection inference. The method is illustrated using surface CO₂ flux measurements obtained from the AmeriFlux network fit with alternative models for the soil CO₂ flux versus soil temperature relationship. The models are fit first to determine pooled parameter estimates across the sites, then using a Bayesian hierarchical method to obtain both global and site-specific parameter estimates. Model comparisons are made using the deviance information criterion (DIC), which considers both goodness of fit and model complexity. The hierarchical models consistently outperform the corresponding pooled models, demonstrating the need for site-specific data and estimates when determining relationships for background soil respiration. A hierarchical model that relates the square root of the CO₂ flux to a quadratic function of soil temperature is found to provide the best fit for the AmeriFlux sites among the models tested. This model also yields effective prediction intervals, consistent with the upper envelope of the flux data across the modeled sites and temperature ranges. Calculation of upper prediction intervals using the proposed method can provide a basis for setting critical values in CO₂ leak detection monitoring at sequestration sites.     

An improved model calculating CO2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar

A thermodynamic model for the solubility of carbon dioxide (CO₂) in pure water and in aqueous NaCl solutions for temperatures from 273 to 533 K, for pressures from 0 to 2000 bar, and for ionic strength from 0 to 4.3 m is presented. The model is based on a specific particle interaction theory for the liquid phase and a highly accurate equation of state for the vapor phase. With this specific interaction approach, this model is able to predict CO₂ solubility in other systems, such as CO₂–H₂O–CaCl₂ and CO₂–seawater, without fitting experimental data from these systems. Comparison of the model predictions with experimental data indicates that the model is within or close to experimental uncertainty, which is about 7% in CO₂ solubility.     

An Uncertainty Quantification Framework for Studying the Effect of Spatial Heterogeneity in Reservoir Permeability on CO2 Sequestration

A new uncertainty quantification framework is adopted for carbon sequestration to evaluate the effect of spatial heterogeneity of reservoir permeability on CO₂ migration. Sequential Gaussian simulation is used to generate multiple realizations of permeability fields with various spatial statistical attributes. In order to deal with the computational difficulties, the following ideas/approaches are integrated. First, different efficient sampling approaches (probabilistic collocation, quasi-Monte Carlo, and adaptive sampling) are used to reduce the number of forward calculations, explore effectively the parameter space, and quantify the input uncertainty. Second, a scalable numerical simulator, extreme-scale Subsurface Transport Over Multiple Phases, is adopted as the forward modeling simulator for CO₂ migration. The framework has the capability to quantify input uncertainty, generate exploratory samples effectively, perform scalable numerical simulations, visualize output uncertainty, and evaluate input-output relationships. The framework is demonstrated with a given CO₂ injection scenario in heterogeneous sandstone reservoirs. Results show that geostatistical parameters for permeability have different impacts on CO₂ plume radius: the mean parameter has positive effects at the top layers, but affects the bottom layers negatively. The variance generally has a positive effect on the plume radius at all layers, particularly at middle layers, where the transport of CO₂ is highly influenced by the subsurface heterogeneity structure. The anisotropy ratio has weak impacts on the plume radius, but affects the shape of the CO₂ plume.     

Prediction of CH4 and CO2 hydrate phase equilibrium and cage occupancy from ab initio intermolecular potentials

Presented is an improved model for the prediction of phase equilibria and cage occupancy of CH₄ and CO₂ hydrate in aqueous systems. Different from most hydrate models that employ Kihara potential or Lennard-Jones potential with parameters derived from experimental phase equilibrium data of hydrates, we use atomic site-site potentials to account for the angle-dependent molecular interactions with parameters directly from ab initio calculation results. Because of this treatment, our model can predict the phase equilibria of CH₄ hydrate and CO₂ hydrate in binary systems over a wide temperature-pressure range (from 243-318 K, and from 10-3000 bar for CH₄ hydrate; from 253-293 K, and from 5-2000 bar for CO₂ hydrate) with accuracy close to experiment. The average deviation of this model from experimental data is less than 3% in pressures for a given temperature. This accuracy is similar to previous models for pressures below 500 bar, but is more accurate than previous models at higher pressures. This model is also capable of predicting the cage occupancy and hydration number for CH₄ hydrate and CO₂ hydrate without fitting any experimental data. The success of this study validates the predictability of ab initio intermolecular potentials for thermodynamic properties.     

Uncertainty in geochemical modelling of CO2 and calcite dissolution in NaCl solutions due to different modelling codes and thermodynamic databases

A prognosis of the geochemical effects of CO₂ storage induced by the injection of CO₂ into geologic reservoirs or by CO₂ leakage into the overlaying formations can be performed by numerical modelling (non-invasive) and field experiments. Until now the research has been focused on the geochemical processes of the CO₂ reacting with the minerals of the storage formation, which mostly consists of quartzitic sandstones. Regarding the safety assessment the reactions between the CO₂ and the overlaying formations in the case of a CO₂ leakage are of equal importance as the reactions in the storage formation. In particular, limestone formations can react very sensitively to CO₂ intrusion. The thermodynamic parameters necessary to model these reactions are not determined explicitly through experiments at the total range of temperature and pressure conditions and are thus extrapolated by the simulation code. The differences in the calculated results lead to different calcite and CO₂ solubilities and can influence the safety issues.This uncertainty study is performed by comparing the computed results, applying the geochemical modelling software codes The Geochemist’s Workbench, EQ3/6, PHREEQC and FactSage/ChemApp and their thermodynamic databases. The input parameters (1) total concentration of the solution, (2) temperature and (3) fugacity are varied within typical values for CO₂ reservoirs, overlaying formations and close-to-surface aquifers. The most sensitive input parameter in the system H₂O–CO₂–NaCl–CaCO₃ for the calculated range of dissolved calcite and CO₂ is the fugacity of CO₂. Hence, the largest range of dissolved calcite is calculated at high fugacities and is 210 mmol/kgw. The average deviation of the results using the databases phreeqc.dat and wateq4f.dat in combination with the code PHREEQC is lowest in comparison to the results of the specific model of Duan and Li, which represents the experimental values at best. Still, the solubility of CO₂ is overestimated in the formation water using these two databases. Therefore, the model results calculate a larger retention capacity, defined as the quantity of CO₂ dissolved in the formation water, than the Duan and Li model would do.     

An integrative approach to robust design and probabilistic risk assessment for CO<Subscript>2</Subscript> storage in geological formations

CO₂ storage in geological formations is currently being discussed intensively as a technology with a high potential for mitigating CO₂ emissions. However, any large-scale application requires a thorough analysis of the potential risks. Current numerical simulation models are too expensive for probabilistic risk analysis or stochastic approaches based on a brute-force approach of repeated simulation. Even single deterministic simulations may require parallel high-performance computing. The multiphase flow processes involved are too non-linear for quasi-linear error propagation and other simplified stochastic tools. As an alternative approach, we propose a massive stochastic model reduction based on the probabilistic collocation method. The model response is projected onto a higher-order orthogonal basis of polynomials to approximate dependence on uncertain parameters (porosity, permeability, etc.) and design parameters (injection rate, depth, etc.). This allows for a non-linear propagation of model uncertainty affecting the predicted risk, ensures fast computation, and provides a powerful tool for combining design variables and uncertain variables into one approach based on an integrative response surface. Thus, the design task of finding optimal injection regimes explicitly includes uncertainty, which leads to robust designs with a minimum failure probability. We validate our proposed stochastic approach by Monte Carlo simulation using a common 3D benchmark problem (Class et al., Comput Geosci 13:451–467, 2009). A reasonable compromise between computational efforts and precision was reached already with second-order polynomials. In our case study, the proposed approach yields a significant computational speed-up by a factor of 100 compared with the Monte Carlo evaluation. We demonstrate that, due to the non-linearity of the flow and transport processes during CO₂ injection, including uncertainty in the analysis leads to a systematic and significant shift of the predicted leakage rates toward higher values compared with deterministic simulations, affecting both risk estimates and the design of injection scenarios.     

Bayesian estimation of reservoir properties—effects of uncertainty quantification of 4D seismic data

This paper shows a history matching workflow with both production and 4D seismic data where the uncertainty of seismic data for history matching comes from Bayesian seismic waveform inversion. We use a synthetic model and perform two seismic surveys, one before start of production and the second after 1 year of production. From the first seismic survey, we estimate the contrast in slowness squared (with uncertainty) and use this estimate to generate an initial estimate of porosity and permeability fields. This ensemble is then updated using the second seismic survey (after inversion to contrasts) and production data with an iterative ensemble smoother. The impact on history matching results from using different uncertainty estimates for the seismic data is investigated. From the Bayesian seismic inversion, we get a covariance matrix for the uncertainty and we compare using the full covariance matrix with using only the diagonal. We also compare with using a simplified uncertainty estimate that does not come from the seismic inversion. The results indicate that it is important not to underestimate the noise in seismic data and that having information about the correlation in the error in seismic data can in some cases improve the results.     

基于高斯过程回归的地下水模型结构不确定性分析与控制

目前针对模型结构不确定性的研究方法主要为贝叶斯模型平均方法,而该方法受到模型权重计算困难等影响,应用受限。基于数据驱动的模型结构误差统计学习方法最近得到关注。研究采用高斯过程回归方法对地下水模型结构误差进行统计模拟,并将DREAMzs算法与高斯过程回归相结合,对地下水模型和统计模型的参数同时进行识别。基于此方法,分别以理想岩溶裂隙海水入侵过程和溶质运移柱体实验为例,进行地下水数值模拟及预测结果的不确定性分析。相对于不考虑模型结构误差条件的不确定性分析,结果表明,考虑结构误差之后,能够明显减少参数识别过程中的参数补偿影响,且能显著提高模型的预测性能。因此,基于高斯过程回归的模型结构不确定性分析可以一定程度控制地下水数值模拟的不确定性,提高模型预测可靠性。     

Post-injection trapping of mobile CO<Subscript><Emphasis Type="Bold">2</Emphasis></Subscript> in deep aquifers: Assessing the importance of model and parameter uncertainties

Predicting the fate of the injected CO₂ is crucial for the safety of carbon storage operations in deep saline aquifers: especially the evolution of the position, the spreading and the quantity of the mobile CO₂ plume during and after the injection has to be understood to prevent any loss of containment. Fluid flow modelling is challenging not only given the uncertainties on subsurface formation intrinsic properties (parameter uncertainty) but also on the modelling choices/assumptions for representing and numerically implementing the processes occurring when CO₂ displaces the native brine (model uncertainty). Sensitivity analysis is needed to identify the group of factors which contributes the most to the uncertainties in the predictions. In this paper, we present an approach for assessing the importance of model and parameter uncertainties regarding post-injection trapping of mobile CO₂. This approach includes the representation of input parameters, the choice of relevant simulation outputs, the assessment of the mobile plume evolution with a flow simulator and the importance ranking for input parameters. A variance-based sensitivity analysis is proposed, associated with the ACOSSO-like meta-modelling technique to tackle the issues linked with the computational burden posed by the use of long-running simulations and with the different types of uncertainties to be accounted for (model and parameter). The approach is tested on a potential site for CO₂ storage in the Paris basin (France) representative of a project in preliminary stage of development. The approach provides physically sound outcomes despite the challenging context of the case study. In addition, these outcomes appear very helpful for prioritizing the future characterisation efforts and monitoring requirements, and for simplifying the modelling exercise.     

Bayesian approach for calibrating transformation model from spatially varied CPT data to regular geotechnical parameter

Cone Penetration Test (CPT) is widely utilized to gain regular geotechnical parameters such as compression modulus, cohesion coefficient and internal friction angle by transformation model in the site investigation. However, it is challenging to obtain simultaneously the unknown coefficients and error of a transformation model, given the intrinsic uncertainty (i.e., spatial variability) of geomaterial and the epistemic uncertainty of geotechnical investigation. A Bayesian approach is therefore proposed calibrating the transformation model based on spatial random field theory. The approach consists of three key elements: (1) three-dimensional anisotropic spatial random field theory; (2) classifications of measurement and error, and the uncertainty propagation diagram of geotechnical investigation; and (3) the unknown coefficients and error calibration of the transformation model given Bayesian inverse modeling method. The massive penetration resistance data from CPT, which is denoted as a spatial random field variable to account for the spatial variability of soil, are classified as type A data. Meanwhile, a few laboratory test data such as the compression modulus are defined as type B data. Based on the above two types of data, the unknown coefficients and error of the transformation model are inversely calibrated with consideration of intrinsic uncertainty of geomaterial, epistemic uncertainties such as measurement errors, prior knowledge uncertainty of transformation model itself, and computing uncertainties of statistical parameters as well as Bayesian method. Baseline studying indicates the proposed approach is applicable to calibrate the transformation model between CPT data and regular geotechnical parameter within spatial random field theory. Next, the calibrated transformation model was compared with classical linear regression in cross-validation, and then it was implemented at three-dimensional site characterization of the background project.     

Determination of lower and upper bounds of predicted production from history-matched models

We present a method to determine lower and upper bounds to the predicted production or any other economic objective from history-matched reservoir models. The method consists of two steps: 1) performing a traditional computer-assisted history match of a reservoir model with the objective to minimize the mismatch between predicted and observed production data through adjusting the grid block permeability values of the model. 2) performing two optimization exercises to minimize and maximize an economic objective over the remaining field life, for a fixed production strategy, by manipulating the same grid block permeabilities, however without significantly changing the mismatch obtained under step 1. This is accomplished through a hierarchical optimization procedure that limits the solution space of a secondary optimization problem to the (approximate) null space of the primary optimization problem. We applied this procedure to two different reservoir models. We performed a history match based on synthetic data, starting from a uniform prior and using a gradient-based minimization procedure. After history matching, minimization and maximization of the net present value (NPV), using a fixed control strategy, were executed as secondary optimization problems by changing the model parameters while staying close to the null space of the primary optimization problem. In other words, we optimized the secondary objective functions, while requiring that optimality of the primary objective (a good history match) was preserved. This method therefore provides a way to quantify the economic consequences of the well-known problem that history matching is a strongly ill-posed problem. We also investigated how this method can be used as a means to assess the cost-effectiveness of acquiring different data types to reduce the uncertainty in the expected NPV.     

Probabilistic parameter estimation and predictive uncertainty based on field measurements for unsaturated soil slope

A key issue in assessment of rainfall-induced slope failure is a reliable evaluation of pore water pressure distribution and its variations during rainstorm, which in turn requires accurate estimation of soil hydraulic parameters. In this study, the uncertainties of soil hydraulic parameters and their effects on slope stability prediction are evaluated, within the Bayesian framework, using the field measured temporal pore-water pressure data. The probabilistic back analysis and parameter uncertainty estimation is conducted using the Markov Chain Monte Carlo simulation. A case study of a natural terrain site is presented to illustrate the proposed method. The 95% total uncertainty bounds for the calibration period are relatively narrow, indicating an overall good performance of the infiltration model for the calibration period. The posterior uncertainty bounds of slope safety factors are much narrower than the prior ones, implying that the reduction of uncertainty in soil hydraulic parameters significantly reduces the uncertainty of slope stability.     

Characterizing uncertainty in geotechnical design of energy piles based on Bayesian theorem

It is known that a lot of uncertainties are involved in geotechnical design of energy piles. In this paper, a Bayesian updating framework is presented to characterize those uncertainties. The load-transfer model is developed to predict the thermomechanical response of energy piles. Considering the cross-case variability of the uncertainty in the axial strains of pile, the global model bias is firstly calibrated by establishing a comprehensive database consisting of 12 energy pile cases. Furthermore, the uncertainty in input parameters is considered in the Bayesian updating of model bias in a specific case. The variability of the uncertain parameters is effectively reduced after updating. The coefficient of variation of prediction is decreased from 0.34 to 0.13. The present framework can well quantify uncertain factors and improve the accuracy and reliability of the prediction model.

     

Percolation-theory and fuzzy rule-based probability estimation of fault leakage at geologic carbon sequestration sites

Leakage of CO₂ and displaced brine from geologic carbon sequestration (GCS) sites into potable groundwater or to the near-surface environment is a primary concern for safety and effectiveness of GCS. The focus of this study is on the estimation of the probability of CO₂ leakage along conduits such as faults and fractures. This probability is controlled by (1) the probability that the CO₂ plume encounters a conductive fault that could serve as a conduit for CO₂ to leak through the sealing formation, and (2) the probability that the conductive fault(s) intersected by the CO₂ plume are connected to other conductive faults in such a way that a connected flow path is formed to allow CO₂ to leak to environmental resources that may be impacted by leakage. This work is designed to fit into the certification framework for geological CO₂ storage, which represents vulnerable resources such as potable groundwater, health and safety, and the near-surface environment as discrete “compartments.” The method we propose for calculating the probability of the network of conduits intersecting the CO₂ plume and one or more compartments includes four steps: (1) assuming that a random network of conduits follows a power-law distribution, a critical conduit density is calculated based on percolation theory; for densities sufficiently smaller than this critical density, the leakage probability is zero; (2) for systems with a conduit density around or above the critical density, we perform a Monte Carlo simulation, generating realizations of conduit networks to determine the leakage probability of the CO₂ plume (P _leak) for different conduit length distributions, densities and CO₂ plume sizes; (3) from the results of Step 2, we construct fuzzy rules to relate P _leak to system characteristics such as system size, CO₂ plume size, and parameters describing conduit length distribution and uncertainty; (4) finally, we determine the CO₂ leakage probability for a given system using fuzzy rules. The method can be extended to apply to brine leakage risk by using the size of the pressure perturbation above some cut-off value as the effective plume size. The proposed method provides a quick way of estimating the probability of CO₂ or brine leaking into a compartment for evaluation of GCS leakage risk. In addition, the proposed method incorporates the uncertainty in the system parameters and provides the uncertainty range of the estimated probability.     

Integrating production data under uncertainty by parallel interacting Markov chains on a reduced dimensional space

In oil industry and subsurface hydrology, geostatistical models are often used to represent the porosity or the permeability field. In history matching of a geostatistical reservoir model, we attempt to find multiple realizations that are conditional to dynamic data and representative of the model uncertainty space. A relevant way to simulate the conditioned realizations is by generating Monte Carlo Markov chains (MCMC). The huge dimensions (number of parameters) of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. In practice, we have to stop the chain far before it has browsed the whole support of the posterior probability density function. Furthermore, as the relationship between the production data and the random field is highly nonlinear, the posterior can be strongly multimodal and the chain may stay stuck in one of the modes. In this work, we propose a methodology to enhance the sampling properties of classical single MCMC in history matching. We first show how to reduce the dimension of the problem by using a truncated Karhunen–Loève expansion of the random field of interest and assess the number of components to be kept. Then, we show how we can improve the mixing properties of MCMC, without increasing the global computational cost, by using parallel interacting Markov Chains. Finally, we show the encouraging results obtained when applying the method to a synthetic history matching case.     

Evaluation of Gaussian approximations for data assimilation in reservoir models

The Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework involves characterizing the posterior distribution of geological parameters in terms of a given prior distribution and data from the reservoir dynamics, together with a forward model connecting the space of geological parameters to the data space. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone. In this paper, we avoid this disadvantage by comparing the ad hoc techniques with a fully resolved state-of-the-art probing of the Bayesian posterior distribution. The ad hoc techniques whose performance we assess are based on (1) linearization around the maximum a posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution, we implement a state-of-the art Markov chain Monte Carlo (MCMC) method that scales well with respect to the dimension of the parameter space, enabling us to study realistic forward models, in two space dimensions, at a high level of grid refinement. Our implementation of the MCMC method provides the gold standard against which the aforementioned Gaussian approximations are assessed. We present numerical synthetic experiments where we quantify the capability of each of the ad hoc Gaussian approximation in reproducing the mean and the variance of the posterior distribution (characterized via MCMC) associated to a data assimilation problem. Both single-phase and two-phase (oil–water) reservoir models are considered so that fundamental differences in the resulting forward operators are highlighted. The main objective of our controlled experiments was to exhibit the substantial discrepancies of the approximation properties of standard ad hoc Gaussian approximations. Numerical investigations of the type we present here will lead to the greater understanding of the cost-efficient, but ad hoc, Bayesian techniques used for data assimilation in petroleum reservoirs and hence ultimately to improved techniques with more accurate uncertainty quantification.     

A Bayesian framework for the validation of models for subsurface flows: synthetic experiments

We present a new Bayesian framework for the validation of models for subsurface flows. We use a compositional model to simulate CO₂ storage in saline aquifers, comparing simulated saturations to observed saturations, together with a Bayesian analysis, to refine the permeability field. At the laboratory scale, we consider a core that is initially fully saturated with brine in a drainage experiment performed at aquifer conditions. Two types of data are incorporated in the framework: the porosity field in the entire core and CO₂ saturation values at equally spaced core slices for several values of time. These parameters are directly measured with a computed tomography scanner. We then find permeability fields that (1) are consistent with the measured parameters and, at the same time, (2) allow one to predict future fluid flow. We combine high performance computing, Bayesian inference, and a Markov chain Monte Carlo (McMC) method for characterizing the posterior distribution of the permeability field conditioned on the available dynamic measurements (saturation values at slices). We assess the quality of our characterization procedure by Monte Carlo predictive simulations, using permeability fields sampled from the posterior distribution. In our characterization step, we solve a compositional two-phase flow model for each permeability proposal and compare the solution of the model with the measured data. To establish the feasibility of the proposed framework, we present computational experiments involving a synthetic permeability field known in detail. The experiments show that the framework captures almost all the information about the heterogeneity of the permeability field of the core. We then apply the framework to real cores, using data measured in the laboratory.