Data-driven surrogates for rapid simulation and optimization of WAG injection in fractured carbonate reservoirs

Conventional simulation of fractured carbonate reservoirs is computationally expensive because of the multiscale heterogeneities and fracture–matrix transfer mechanisms that must be taken into account using numerical transfer functions and/or detailed models with a large number of simulation grid cells. The computational requirement increases significantly when multiple simulation runs are required for sensitivity analysis, uncertainty quantification and optimization. This can be prohibitive, especially for giant carbonate reservoirs. Yet, sensitivity analysis, uncertainty quantification and optimization are particularly important to analyse, determine and rank the impact of geological and engineering parameters on the economics and sustainability of different Enhanced Oil Recovery (EOR) techniques. We use experimental design to set up multiple simulations of a high-resolution model of a Jurassic carbonate ramp, which is an analogue for the highly prolific reservoirs of the Arab D Formation in Qatar. We consider CO2 water-alternating-gas (WAG) injection, which is a successful EOR method for carbonate reservoirs. The simulations are employed as a basis for generating datadriven surrogate models using polynomial regression and polynomial chaos expansion. Furthermore, the surrogates are validated by comparing surrogate predictions with results from numerical simulation and estimating goodness-of-fit measures. In the current work, parameter uncertainties affecting WAG modelling in fractured carbonates are evaluated, including fracture network properties, wettability and fault transmissibility. The results enable us to adequately explore the parameter space, and to quantify and rank the interrelated effect of uncertain model parameters on CO2 WAG efficiency. The results highlight the first order impact of the fracture network properties and wettability on hydrocarbon recovery and CO2 utilization during WAG injection. In addition, the surrogate models enable us to calculate quick estimates of probabilistic uncertainty and to rapidly optimize WAG injection, while achieving significant computational speed-up compared with the conventional simulation framework

Opportunities and challenges in CO2 geologic utilization and storage

CO2 geological utilization and storage is considered as an effective approach to deeply cut anthropogenic CO2 emissions. It is vital to enhance the amount of CO2 stored in the subsurface, at the same time to ensure safe and long-term subsurface storage of CO2 without any CO2 leakage. Science and engineering research in modeling concepts, experimental approaches, safety assurance and emerging CO2 geological utilization and storage technologies have driven the advancement of CO2 geological utilization and storage in recent years. In order to encourage communication and collaboration in CO2 geological utilization and storage research worldwide, a Sino-German joint symposium titled “Opportunities and Challenges in CO2 Geologic Utilization and Storage” was organized in Wuhan and Stuttgart from February 22 to 24, 2023, bringing together experts from China, Germany, and other countries. The symposium was jointly organized by Institute of Rock and Soil Mechanics, Chinese Academy of Sciences and Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart with financial support from the Sino-German Center for Research Promotion. A two-site hybrid meeting was held (participants in China met in Wuhan, participants in Germany met in Stuttgart, and other participants joined the meeting online), attracting more than 100 participants from around the world. The latest studies in the field of CO2 geological utilization and storage were presented at the symposium.Cited as: Zhang, L., Nowak, W., Oladyshkin, S., Wang, Y., Cai, J. Opportunities and challenges in CO2 geologic utilization and storage. Advances in Geo-Energy Research, 2022, 8(3): 141-145. https://doi.org/10.46690/ager.2023.06.0

Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond

Uncertainty-aware Validation Benchmarks for Coupling Free Flow and Porous-Medium Flow

A correct choice of interface conditions and useful model parameters for coupled free-flow and porous-medium systems is vital for physically consistent modeling and accurate numerical simulations of applications. We consider the Stokes--Darcy problem with different models for the porous-medium compartment and corresponding coupling strategies: the standard averaged model based on Darcy's law with classical or generalized interface conditions, as well as the pore-network model. We study the coupled flow problems' behaviors considering a benchmark case where a pore-scale resolved model provides the reference solution and quantify the uncertainties in the models' parameters and the reference data. To achieve this, we apply a statistical framework that incorporates a probabilistic modeling technique using a fully Bayesian approach. A Bayesian perspective on a validation task yields an optimal bias-variance trade-off against the reference data. It provides an integrative metric for model validation that incorporates parameter and conceptual uncertainty. Additionally, a model reduction technique, namely Bayesian Sparse Polynomial Chaos Expansion, is employed to accelerate the calibration and validation processes for computationally demanding Stokes--Darcy models with different coupling strategies. We perform uncertainty-aware validation, demonstrate each model's predictive capabilities, and make a model comparison using a Bayesian validation metric

Surrogate-based Bayesian Comparison of Computationally Expensive Models: Application to Microbially Induced Calcite Precipitation

Geochemical processes in subsurface reservoirs affected by microbial activity change the material properties of porous media. This is a complex biogeochemical process in subsurface reservoirs that currently contains strong conceptual uncertainty. This means, several modeling approaches describing the biogeochemical process are plausible and modelers face the uncertainty of choosing the most appropriate one. Once observation data becomes available, a rigorous Bayesian model selection accompanied by a Bayesian model justifiability analysis could be employed to choose the most appropriate model, i.e. the one that describes the underlying physical processes best in the light of the available data. However, biogeochemical modeling is computationally very demanding because it conceptualizes different phases, biomass dynamics, geochemistry, precipitation and dissolution in porous media. Therefore, the Bayesian framework cannot be based directly on the full computational models as this would require too many expensive model evaluations. To circumvent this problem, we suggest performing both Bayesian model selection and justifiability analysis after constructing surrogates for the competing biogeochemical models. Here, we use the arbitrary polynomial chaos expansion. We account for the approximation error in the Bayesian analysis by introducing novel correction factors for the resulting model weights. Thereby, we extend the Bayesian justifiability analysis and assess model similarities for computationally expensive models. We demonstrate the method on a representative scenario for microbially induced calcite precipitation in a porous medium. Our extension of the justifiability analysis provides a suitable approach for the comparison of computationally demanding models and gives an insight on the necessary amount of data for a reliable model performance

Découplage de la thermodynamique et hydrodynamique et solutions asymptotiques des problèmes d'écoulement compositionnel gaz-liquide en milieux poreux

The present work deals with the problem of the compositional gas-liquid flow for the well representation in reservoir simulations. The objective is to develop analytical relationships which would be able to link the wellbore pressure, saturation and component concentrations to their mean values within each zone of the well influence. It is shown that N-2 equations describing the transport of phase concentrations can be transformed into the space- and time-independent ordinary differential equations (differentiation with respect to pressure) when examined along flow streamlines. These transformed equations represent additional thermodynamic relations reducing the thermodynamic degree of freedom. Due to this the thermodynamic variance of the limit compositional model is shown to be equal to 1 for any number of chemical components. This transformation ensure a total splitting of the limit compositional model into the new thermodynamic model and a hydrodynamic model, which may be resoved inedpendently of one another. The split thermodynamic model is totally independent on the hydrodynamic one, and describes the equilibrium behaviour of an open gas-liquid system. This model contains the classic equilibrium equations and EOS, as well as N-2 new differential equations called the "delta-law" which determine the composition variation in an open system, in which the mass of each component is not conserved. The split hydrodynamic model consists of two equations for pressure and saturation. The split hydrodynamic model was used to develop asymptotic solutions of gas-condensate flow problems. The problem was shown to be singularly perturbed with formation of a boundary layer in the vicinity of the well. In this layer the basic contrast property of gas and liquid mobilities is perturbed. A special technique is developed which enables to construct asymptotic expansions in the form of two various series, one of them is valid far from the well (the exterior expansion), while the second one in valid in the vicinity of the well (the boundary-layer or interior expansion). By applying the suggested asymptotic method, we have developed the asymptotic solutions for the problem of multi-component gas-condensate flow to a well in a bounded domain at a variable flow rate. In several cases the solution may be obtained in the analytical form, while in general case of flow the method leads to a semi-analytical solution presented as an initial problem for a differential equation. This solution, even being presented in non-analytical form, is much simpler than the original compositional model, as the equation for saturation does not depend on the local pressure, but on the boundary pressure only. In the last chapter we extended this approach to the case when the capillary pressure is not neglected. We assumed however that the capillary forces are lower than the pressure difference between the wellbore and reservoir boundary, due to which we applied the perturbation method over the small inverse capillary number. The improved asymptotic solutions are obtained which take into account the capillary effect. Numerical simulations shown that these effects are maximal in the vicinity of the well. For the practice, the obtained asymptotic solutions may be used in the following way to resolve the problem of gas-condensate well representation. The case of a long-term exploitation of the reservoir}. First of all, the traditional simulation of the reservoir behaviour can be performed with ECLIPSE by adding the Peaceman method of well representation, which is an analytical relation for the wellbore pressure via the production rate. This relation includes a condensate saturation which can be evaluated as a mean reservoir saturation. Such a simulation provides a good result for the wellbore pressure (or the production rate), and a good result for the boundary saturation, but poor data for the wellbore saturation. This value can be calculated next by using the asymptotic solutions suggested in the presented project. The case of a short-term well production (a well test). It is sufficient to simulate the reservoir behaviour in the domain of the well influence, by assuming that the boundary state remains invariable (and known a priori). In this case the asymptotic solutions suggested in the presnet work can be directly used to simulate the problem (without using ECLIPSE)Le travail actuel traite le problème de l'écoulement gaz-liquide compositionnel pour la représentation d'un puits dans des simulations de réservoir. L'objectif est de développer les rapports analytiques qui pourraient lier la pression de puits, la saturation et les concentrations de composant à leurs valeurs moyennes dans chaque zone de l'influence de puits. Nous avons montre que des N-2 équations décrivant le transport des concentrations de phase peuvent être transformées en équations ordinaires (différentiation en ce qui concerne la pression) indépendantes du temps et de l'espace examinant le long des lignes de courant. Ces équations transformées représentent des relations thermodynamiques additionnelles réduisant le degré de liberté thermodynamique. En raison de ceci le variance thermodynamique du modèle compositionnel limite s'avère égal à 1 pour tout nombre de composants chimiques. Cette transformation assurent se découplage total du modèle compositionnel limite dans le nouveau modèle thermodynamique et le modèle hydrodynamique, qui peut être resoved inedpendently d'un un autre. Le modèle thermodynamique décompose est totalement indépendant sur l'hydrodynamique, et décrit le comportement d'équilibre d'un système gaz-liquide ouvert. Ce modèle contient les équations d'équilibre et la EOS classiques, aussi bien que les N-2 nouvelles équations appelées la "Delta-loi", qui déterminent la variation de composition d'un système ouvert dans lequel la masse de chaque composant n'est pas conservée. Le modèle hydrodynamique décompose a été utiliser pour développer les solutions asymptotiques des problèmes d'écoulement de gaz-condensat. Le problème a été montré perturbé singulièrement avec la formation d'une couche limite à voisinage du puits. Dans cette couche la propriété basique de contraste des mobilities de gaz et de liquide est perturbée. On développe une technique spéciale qui permet de construire des expansions asymptotiques sous forme de deux diverses séries: le primer est valide loin du puits (l'expansion extérieure), alors que le second dans valide à voisinage du puits (la couche limite ou l'expansion intérieure). En appliquant la méthode asymptotique suggérée, nous avons développé les solutions asymptotiques pour le problème de l'écoulement multicompositionnel de gaz-condensat àu puits dans un domaine borné à un débit variable. En plusieurs cas la solution peut être obtenue sous la forme analytique, alors qu'en cas général de l'écoulement la méthode mène à une solution de semi-analytical, présentée comme problème initial pour une équation. Cette solution, même étant présenté en forme non-analytique, est beaucoup plus simple que le modèle compositionnel original, car l'équation pour la saturation ne dépend pas de la pression locale, mais dépend de la pression de bord seulement. Dans le dernier chapitre nous avons prolongé cette approche au cas quand la pression capillaire n'est pas négligée. Nous avons supposé cependant que les forces capillaires sont inférieures à la différence de pression entre le puits et la bord de réservoir, dus à ce que nous avons appliqué la méthode de perturbation pour petit nombre capillaire inverse. On obtient les solutions asymptotiques améliorées qui tiennent compte de l'effet capillaire. Simulations numériques montrées que ces effets sont maximaux àu voisinage du puits. Le cas d'une exploitation à long terme du réservoir. Tout d'abord, la simulation traditionnelle du comportement de réservoir peut être effectuée avec l'ECLIPSE en ajoutant la méthode de Peaceman de représentation bonne, qui est une relation analytique pour la pression de puits par l'intermédiaire du débit de production. Cette relation inclut une saturation condensat qui peut être évaluée comme saturation moyenne de réservoir. Une telle simulation fournit un bon résultat pour la pression de puits (ou le débit de production), et un bon résultat pour la saturation de bord, mais des données faibles pour la saturation de puits. Cette valeur peut être calculée en utilisant les solutions asymptotiques suggérées dans le présent projet. Le cas d'un puits de production à court terme (un essai de puits). Il est suffisant de simuler le comportement de réservoir dans le domaine de l'influence de puits, en supposant que l'état de frontière demeure invariable (et connu a priori). Dans ce cas-ci les solutions asymptotiques suggérées dans le travail de presnet peuvent être directement employées pour simuler le problème (sans employer l'ECLIPSE). Le problème de l'écoulement de gaz-condensat à une fracture. Nous avons construit un champ plutôt arbitraire avec des lignes de courant orientées à la fracture, en supposant que la fracture joue le rôle d'une décharge, et les lignes de courant sont stationnaire. Pour une ligne de courant arbitraire nous avons reformulé le modèle d'écoulement de gaz-condensat dans des coordonnées cartésiennes. Pour ce problème nous avons développé les expansions asymptotique