Seal integrity and feasibility of CO<Subscript>2</Subscript> sequestration in the Teapot Dome EOR pilot: geomechanical site characterization

This paper reports a preliminary investigation of CO₂ sequestration and seal integrity at Teapot Dome oil field, Wyoming, USA, with the objective of predicting the potential risk of CO₂ leakage along reservoir-bounding faults. CO₂ injection into reservoirs creates anomalously high pore pressure at the top of the reservoir that could potentially hydraulically fracture the caprock or trigger slip on reservoir-bounding faults. The Tensleep Formation, a Pennsylvanian age eolian sandstone is evaluated as the target horizon for a pilot CO₂ EOR-carbon storage experiment, in a three-way closure trap against a bounding fault, termed the S1 fault. A preliminary geomechanical model of the Tensleep Formation has been developed to evaluate the potential for CO₂ injection inducing slip on the S1 fault and thus threatening seal integrity. Uncertainties in the stress tensor and fault geometry have been incorporated into the analysis using Monte Carlo simulation. The authors find that even the most pessimistic risk scenario would require ∼10 MPa of excess pressure to cause the S1 fault to reactivate and provide a potential leakage pathway. This would correspond to a CO₂ column height of ∼1,500 m, whereas the structural closure of the Tensleep Formation in the pilot injection area does not exceed 100 m. It is therefore apparent that CO₂ injection is not likely to compromise the S1 fault stability. Better constraint of the least principal stress is needed to establish a more reliable estimate of the maximum reservoir pressure required to hydrofracture the caprock.     

A stochastic inversion workflow for monitoring the distribution of CO<Subscript>2</Subscript> injected into deep saline aquifers

We present a two-step stochastic inversion approach for monitoring the distribution of CO₂ injected into deep saline aquifers for the typical scenario of one single injection well and a database comprising a common suite of well logs as well as time-lapse vertical seismic profiling (VSP) data. In the first step, we compute several sets of stochastic models of the elastic properties using conventional sequential Gaussian co-simulations (SGCS) representing the considered reservoir before CO₂ injection. All realizations within a set of models are then iteratively combined using a modified gradual deformation algorithm aiming at reducing the mismatch between the observed and simulated VSP data. In the second step, these optimal static models then serve as input for a history matching approach using the same modified gradual deformation algorithm for minimizing the mismatch between the observed and simulated VSP data following the injection of CO₂. At each gradual deformation step, the injection and migration of CO₂ is simulated and the corresponding seismic traces are computed and compared with the observed ones. The proposed stochastic inversion approach has been tested for a realistic, and arguably particularly challenging, synthetic case study mimicking the geological environment of a potential CO₂ injection site in the Cambrian-Ordivician sedimentary sequence of the St. Lawrence platform in Southern Québec. The results demonstrate that the proposed two-step reservoir characterization approach is capable of adequately resolving and monitoring the distribution of the injected CO₂. This finds its expression in optimized models of P- and S-wave velocities, density, and porosity, which, compared to conventional stochastic reservoir models, exhibit a significantly improved structural similarity with regard to the corresponding reference models. The proposed approach is therefore expected to allow for an optimal injection forecast by using a quantitative assimilation of all available data from the appraisal stage of a CO₂ injection site.     

Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

Stochastic design of beams on reinforced random earth beds in deterministic mode

One of the main soil parameters in analysis and design of foundations is modulus of subgrade reaction (MSR) which is a stochastic process. However, design engineers prefer a deterministic approach invoking mean of MSR and rather empirical factors of safety to account for the uncertainty. The present study includes the stochasticity in the deterministic designs by linking the factors of safety (in respect of maximum deflection and bending moment) to the allowable risk of failure through a Monte Carlo simulation on a lumped parameter deterministic model. A parametric study reveals that for a given risk level, the factors of safety are strongly dependent upon the coefficient of variation of MSR, and only mildly upon other geometric parameters of foundation system. This facilitates development of closed form equations for the upper bounds on factors of safety exclusively in terms of allowable risk of failure and the coefficient of variation of MSR.     

The inclusive and simplified forms of Bayesian interpolation for general and monotonic models using Gaussian and Generalized Beta distributions with application to Monte Carlo simulations

A recently developed Bayesian interpolation method (BI) and its application to safety assessment of a flood defense structure are described in this paper. We use a one-dimensional Bayesian Monte Carlo method (BMC) that has been proposed in (Rajabalinejad 2009) to develop a weighted logical dependence between neighboring points. The concept of global uncertainty is adequately explained and different uncertainty association models (UAMs) are presented for linking the local and global uncertainty. Based on the global uncertainty, a simplified approach is introduced. By applying the global uncertainty, we apply the Guassian error estimation to general models and the Generalized Beta (GB) distribution to monotonic models. Our main objective in this research is to simplify the newly developed BMC method and demonstrate that it can dramatically improve the simulation efficiency by using prior information from outcomes of the preceding simulations. We provide theory and numerical algorithms for the BI method geared to multi-dimensional problems, integrate it with a probabilistic finite element model, and apply the coupled models to the reliability assessment of a flood defense for the 17th Street Flood Wall system in New Orleans.     

Implications of spatial reservoir uncertainty for CO2 sequestration in the east Snake River Plain, Idaho (USA)

Basalt-hosted hydrogeologic systems have been proposed for geologic CO₂ sequestration based on laboratory research suggesting rapid mineralization rates. However, despite this theoretical appeal, little is known about the impacts of basalt fracture heterogeneity on CO₂ migration at commercial scales. Evaluating the suitability of basalt reservoirs is complicated by incomplete knowledge of in-situ fracture distributions at depths required for CO₂ sequestration. In this work, a numerical experiment is used to investigate the effects of spatial reservoir uncertainty for geologic CO₂ sequestration in the east Snake River Plain, Idaho (USA). Two criteria are investigated: (1) formation injectivity and (2) confinement potential. Several theoretical tools are invoked to develop a field-based approach for geostatistical reservoir characterization and their implementation is illustrated. Geologic CO₂ sequestration is simulated for 10?years of constant-rate injection at ~680,000 tons per year and modeled by Monte Carlo simulation such that model variability is a function of spatial reservoir heterogeneity. Results suggest that the spatial distribution of heterogeneous permeability structures is a controlling influence on formation injectivity. Analysis of confinement potential is less conclusive; however, in the absence of confining sedimentary interbeds within the basalt pile, rapid mineralization may be necessary to reduce the risk of escape.     

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.     

An approximate dynamic programming approachto decision making in the presence of uncertainty for surfactant-polymer flooding

The least squares Monte Carlo method is a decision evaluation method that can capture the effect of uncertainty and the value of flexibility of a process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on the Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations. The objective of this study was to enhance the performance of the least squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in obtaining the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus, we introduce the least squares probabilistic collocation method. The decision evaluation considered a number of technical and economic uncertainties. Three reservoir case studies were used: a simple homogeneous model, the PUNQ-S3 model, and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method. Different possible enhancements were discussed in order to practically adapt the least squares probabilistic collocation method to more realistic and complex reservoir models. Furthermore, it is desired to perform the method to evaluate high-dimensional decision scenarios for different chemical enhanced oil recovery processes using real reservoir data.     

An adaptive sampling approach to reduce uncertainty in slope stability analysis

An adaptive sampling approach is proposed, which can sample spatially varying shear strength parameters efficiently to reduce uncertainty in the slope stability analysis. This approach employs a limit equilibrium model and stochastic conditional methodology to determine the likely sampling locations. Karhunen-Loève expansion is used to conduct the conditional Monte Carlo simulation. A first-order analysis is also proposed to ease the computational burden associated with Monte Carlo simulation. These approaches are then tested using borehole data from a field site. Results indicate that the proposed adaptive sampling approach is an effective and efficient sampling scheme for reducing uncertainty in slope stability analysis.     

A Probabilistic Modelling System for Assessing Flood Risks

In order to be economically viable, flood disaster mitigation should be based on a comprehensive assessment of the flood risk. This requires the estimation of the flood hazard (i.e. runoff and associated probability) and the consequences of flooding (i.e. property damage, damage to persons, etc.). Within the “German Research Network Natural Disasters” project, the working group on “Flood Risk Analysis” investigated the complete flood disaster chain from the triggering event down to its various consequences. The working group developed complex, spatially distributed models representing the relevant meteorological, hydrological, hydraulic, geo-technical, and socio-economic processes. In order to assess flood risk these complex deterministic models were complemented by a simple probabilistic model. The latter model consists of modules each representing one process of the flood disaster chain. Each module is a simple parameterisation of the corresponding more complex model. This ensures that the two approaches (simple probabilistic and complex deterministic) are compatible at all steps of the flood disaster chain. The simple stochastic approach allows a large number of simulation runs in a Monte Carlo framework thus providing the basis for a probabilistic risk assessment. Using the proposed model, the flood risk including an estimation of the flood damage was quantified for an example area at the river Rhine. Additionally, the important influence of upstream levee breaches on the flood risk at the lower reaches was assessed. The proposed model concept is useful for the integrated assessment of flood risks in flood prone areas, for cost-benefit assessment and risk-based design of flood protection measures and as a decision support tool for flood management.     

Robust quantification of parametric uncertainty for surfactant–polymer flooding

Uncertainty in surfactant–polymer flooding is an important challenge to the wide-scale implementation of this process. Any successful design of this enhanced oil recovery process will necessitate a good understanding of uncertainty. Thus, it is essential to have the ability to quantify this uncertainty in an efficient manner. Monte Carlo simulation is the traditional uncertainty quantification approach that is used for quantifying parametric uncertainty. However, the convergence of Monte Carlo simulation is relatively low, requiring a large number of realizations to converge. This study proposes the use of the probabilistic collocation method in parametric uncertainty quantification for surfactant–polymer flooding using four synthetic reservoir models. Four sources of uncertainty were considered: the chemical flood residual oil saturation, surfactant and polymer adsorption, and the polymer viscosity multiplier. The output parameter approximated is the recovery factor. The output metrics were the input–output model response relationship, the probability density function, and the first two moments. These were compared with the results obtained from Monte Carlo simulation over a large number of realizations. Two methods for solving for the coefficients of the output parameter polynomial chaos expansion are compared: Gaussian quadrature and linear regression. The linear regression approach used two types of sampling: full-tensor product nodes and Chebyshev-derived nodes. In general, the probabilistic collocation method was applied successfully to quantify the uncertainty in the recovery factor. Applying the method using the Gaussian quadrature produced more accurate results compared with using the linear regression with full-tensor product nodes. Applying the method using the linear regression with Chebyshev derived sampling also performed relatively well. Possible enhancements to improve the performance of the probabilistic collocation method were discussed. These enhancements include improved sparse sampling, approximation order-independent sampling, and using arbitrary random input distribution that could be more representative of reality.     

Reliability analysis of soil liquefaction based on standard penetration test

A review of probabilistic and deterministic liquefaction evaluation procedures reveals that there is a need for a comprehensive approach that accounts for different sources of uncertainty in liquefaction evaluations. For the same set of input parameters, different models provide different factors of safety and/or probabilities of liquefaction. To account for the different uncertainties, including both the model and measurement uncertainties, reliability analysis is necessary. This paper presents a review and comparative study of such reliability approaches that can be used to obtain the probability of liquefaction and the corresponding factor of safety. Using a simplified deterministic Seed method, this reliability analysis has been performed. The probability of liquefaction along with the corresponding factor of safety have been determined based on a first order second moment (FOSM) method, an advanced FOSM (Hasofer–Lind) reliability method, a point estimation method (PEM) and a Monte Carlo simulation (MCS) method. A combined method that uses both FOSM and PEM is presented and found to be simple and reliable for liquefaction analysis. Based on the FOSM reliability approach, the minimum safety factor value to be adopted for soil liquefaction analysis (depending on the variability of soil resistance, shear stress parameters and acceptable risk) has been studied and a new design safety factor based on a reliability approach is proposed.     

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.     

Screening and ranking framework for geologic CO<Subscript>2</Subscript> storage site selection on the basis of health,safety, and environmental risk

A screening and ranking framework (SRF) has been developed to evaluate potential geologic carbon dioxide (CO₂) storage sites on the basis of health, safety, and environmental (HSE) risk arising from CO₂ leakage. The approach is based on the assumption that CO₂ leakage risk is dependent on three basic characteristics of a geologic CO₂ storage site: (1) the potential for primary containment by the target formation; (2) the potential for secondary containment if the primary formation leaks; and (3) the potential for attenuation and dispersion of leaking CO₂ if the primary formation leaks and secondary containment fails. The framework is implemented in a spreadsheet in which users enter numerical scores representing expert opinions or published information along with estimates of uncertainty. Applications to three sites in California demonstrate the approach. Refinements and extensions are possible through the use of more detailed data or model results in place of property proxies.     

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.     

Probabilistic time planning for underground mines

Summary The advantages of probabilistic time planning techniques, compared to deterministic ones, are discussed and an approach to probabilistic planning is presented. The approach includes an analysis of disturbancy factors, and a method of estimating the distribution of project completion time by using Monte Carlo simulation. It is shown how this result may be used to calculate the need for development buffers, which also has been demonstrated on a particular case, the Oscar project, a sublevel stoping operation in the Kiruna mine, Sweden.     

Reliability analysis of serviceability performance for an underground cavern using a non-intrusive stochastic method

This paper proposes a non-intrusive stochastic analysis procedure for reliability analysis of the serviceability performance of an underground cavern with an implicit limit state function. This procedure is formulated on the basis of the stochastic response surface method (SRSM) and the deterministic finite element method. First, the SRSM is briefly introduced and implemented through a MATLAB code. Then, the software SIGMA/W is used to perform a deterministic finite element analysis. Next, a link between the MATLAB code and SIGMA/W is developed to automatically pass exchange data between the two platforms. Finally, two examples are presented to illustrate the capacity and validity of the proposed procedure. In the first example, a closed-form limit state function is adopted to validate the SRSM by comparing it with the results obtained from a direct Monte Carlo simulation. In the second example, the serviceability performance of an underground cavern is analyzed to illustrate the capacity of the proposed procedure to handle a reliability problem with an implicit limit state function. The proposed procedure does not require the user to modify the existing deterministic finite element code. The deterministic finite element analysis and the probabilistic analysis are decoupled. This is a major practical advantage because realistic probabilistic analyses are made possible. The SRSM can produce sufficiently accurate reliability results. Furthermore, the method is much more efficient than the direct Monte Carlo simulation. Sensitivity analyses show the effect of the variability of input random variables and the correlation between them on: (1) the probability density functions, (2) the first four order statistical moments, and (3) the probability of failure, which is investigated and discussed.     

Reliability analysis of drilled shaft behavior using finite difference method and <Emphasis Type="Italic">Monte Carlo</Emphasis> simulation

Load displacement analysis of drilled shafts can be accomplished by utilizing the “t-z” method, which models soil resistance along the length and tip of the drilled shaft as a series of springs. For non-linear soil springs, the governing differential equation that describes the soil-structure interaction may be discretized into a set of algebraic equations based upon finite difference methods. This system of algebraic equations may be solved to determine the load–displacement behavior of the drilled shaft when subjected to compression or pullout. By combining the finite difference method with Monte Carlo simulation techniques, a probabilistic load–displacement analysis can be conducted. The probabilistic analysis is advantageous compared to standard factor of safety design because uncertainties with the shaft–soil interface and tip properties can be independently quantified. This paper presents a reliability analysis of drilled shaft behavior by combining the finite difference technique for analyzing non-linear load–displacement behavior with Monte Carlo simulation method. As a result we develop probabilistic relationships for drilled shaft design for both total stress (undrained) and effective stress (drained) parameters. The results are presented in the form of factor of safety or resistance factors suitable for serviceability design of drilled shafts.     

Thermodynamics of mixing in diopside–jadeite,CaMgSi<Subscript>2</Subscript>O<Subscript>6</Subscript>–NaAlSi<Subscript>2</Subscript>O<Subscript>6</Subscript>, solid solution from static lattice energy calculations

Static lattice energy calculations (SLEC), based on empirical interatomic potentials, have been performed for a set of 800 different structures in a 2 × 2 × 4 supercell of C2/c diopside with compositions between diopside and jadeite, and with different states of order of the exchangeable Na/Ca and Mg/Al cations. Excess static energies of these structures have been cluster expanded in a basis set of 37 pair-interaction parameters. These parameters have been used to constrain Monte Carlo simulations of temperature-dependent properties in the range of 273–2,023 K and to calculate a temperature–composition phase diagram. The simulations predict the order–disorder transition in omphacite at 1,150 ± 20°C in good agreement with the experimental data of Carpenter (Mineral Petrol 78:433–440, 1981). The stronger ordering of Mg/Al within the M1 site than of Ca/Na in the M2 site is attributed to the shorter M1–M1 nearest-neighbor distance, and, consequently, the stronger ordering force. The comparison of the simulated relationship between the order parameters corresponding to M1 and M2 sites with the X-ray refinement data on natural omphacites (Boffa Ballaran et al. in Am Mineral 83:419–433, 1998) suggests that the cation ordering becomes kinetically ineffective at about 600°C.     

结合全球气候模型预测结果概率分析融化深度

A probabilistic approach may be adopted to predict freeze and thaw depths to account for the variability of (1) material properties, and (2) contemporary and future surface energy input parameters (e.g.air temperatures, cloud cover, snow cover) predicted with global climate models. To illustratc the probabilistic approach, an example of the prediction of thaw depths in Fairbanks, Alaska, is considered, More specifically, the Stefan equation is used together with the Monte Carlo simulation technique to make a probabilistic prediction of thaw penetration. The simulation results indicate that the variability in material properties, surface energy input parameters, and temperature data can lead to significant uncertainty in predicting thaw penetration. The Taylor series method was performed to determine the mean and standard deviation of thaw penetration and the results were compared to the Monte Carlo simulation results. The close comparison of the results suggests that the simpler Taylor series method may be applied to many cold regions problems to account for the variability of input parameters.