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Abstract: Abstract A three dimensional parallel implementation of Multiscale Mixed Methods based on non-overlapping domain decomposition techniques is proposed for multi-core computers and its computational performance is assessed by means of numerical experiments. As a prototypical method, from which many others can be derived, the Multiscale Robin Coupled Method is chosen and its implementation explained in detail. Numerical results for problems ranging from millions up to more than 2 billion computational cells in highly heterogeneous anisotropic rock formations based on the SPE10 benchmark are shown. The proposed implementation relies on direct solvers for both local problems and the interface coupling system. We find good weak and strong scalalability as compared against a state-of-the-art global fine grid solver based on Algebraic Multigrid preconditioning in single and two-phase flow problems. PubDate: 2022-06-01
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Abstract: Abstract In this work we present an efficient implementation of Eulerian TVD methods. We apply parallelization strategies based entirely on GPU for the solution of the 2D transport equation in heterogeneous porous media. Additionally, a parallel strategy is proposed for the generation of exponentially correlated lognormally distributed permeability fields in GPU. The programs are developed using C++/CUDA. The implemented methods are used to solve advective dominant problems, in a context of Monte Carlo type simulations to numerically determine the longitudinal and transversal macrodispersion coefficients averaging over 100 simulations for permeability fields for a large range of variances. The following types of transport are considered for testing: pure advection, advection-diffusion and advection-dispersion. The performance in terms of the computation time of explicit and implicit methods are compared. We show that the implemented algorithms allow to efficiently solve problems in computational domains of up to 134.5 million cells in a single GPU. PubDate: 2022-06-01
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Abstract: Abstract Traditional stochastic algorithms for characterizing fracture networks are purely based on statistical inferences from outcrop images, and therefore the models produced, may not be physically realistic because they may not be consistent with the process of propagation and termination of fractures. These processes are better represented in geomechanical models of the fracturing process. However, full-physics numerical models are computationally inefficient for modeling fractures at a reservoir scale while accounting for material heterogeneities. More importantly, geomechanical simulations yield deterministic results, thus failing to represent the inherent uncertainties due to input properties and paleo stress conditions. In order to facilitate geomechanical characterization, in this research, a number of small-scale, high fidelity, finite discrete element method (FDEM) based forward models are executed and the relationship between prevailing stress conditions and the fracture propagation direction is inferred using Machine Learning (ML) approaches. We develop a ML based fracture network modeling approach that is orders of magnitude faster, efficiently scalable and may extend the capabilities of statistical fracture modeling approaches by accounting for the physical process of fracture propagation and uncertainties associated with geomechanical parameters. The application and effectiveness of this ML based modeling approach is demonstrated using a synthetic case and a case study from Teapot Dome, Wyoming based on the fracture characteristics inferred from the FMI logs near well 67-1-x-10 in the Tensleep Formation reported by Schwartz (2006). PubDate: 2022-06-01
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Abstract: Abstract Ensemble Kalman filters are based on a Gaussian assumption, which can limit their performance in some non-Gaussian settings. This paper reviews two nonlinear, non-Gaussian extensions of the Ensemble Kalman Filter: Gaussian anamorphosis (GA) methods and two-step updates, of which the rank histogram filter (RHF) is a prototypical example. GA-EnKF methods apply univariate transforms to the state and observation variables to make their distribution more Gaussian before applying an EnKF. The two-step methods use a scalar Bayesian update for the first step, followed by linear regression for the second step. The connection of the two-step framework to the full Bayesian problem is made, which opens the door to more advanced two-step methods in the full Bayesian setting. A new method for the first part of the two-step framework is proposed, with a similar form to the RHF but a different motivation, called the ‘improved RHF’ (iRHF). A suite of experiments with the Lorenz-‘96 model demonstrate situations where the GA-EnKF methods are similar to EnKF, and where they outperform EnKF. The experiments also strongly support the accuracy of the RHF and iRHF filters for nonlinear and non-Gaussian observations; these methods uniformly beat the EnKF and GA-EnKF methods in the experiments reported here. The new iRHF method is only more accurate than RHF at small ensemble sizes in the experiments reported here. PubDate: 2022-06-01
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Abstract: Abstract This work investigates an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints in reservoir data assimilation problems. The proposed workflow is built upon a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES), and inherits the benefits of ensemble-based data assimilation algorithms in geoscience applications. Unlike the traditional ensemble assimilation algorithms, the proposed workflow admits cost functions beyond the form of nonlinear-least-squares, and has the potential to develop an infinite number of constrained assimilation algorithms. In the proposed workflow, we treat data assimilation with constraints as a constrained optimization problem. Instead of relying on a general-purpose numerical optimization algorithm to solve the constrained optimization problem, we derive an (approximate) closed form to iteratively update model variables, but without the need to explicitly linearize the constraint systems. The established model update formula bears similarities to that of an iterative ensemble smoother (IES). Therefore, in terms of theoretical analysis, it becomes relatively easy to transit from an ordinary IES to the proposed constrained assimilation algorithms, and in terms of practical implementation, it is also relatively straightforward to implement the proposed workflow for users who are familiar with the IES, or other conventional ensemble data assimilation algorithms like the ensemble Kalman filter (EnKF). Apart from the aforementioned features, we also develop efficient methods to handle two noticed issues that would be of practical importance for ensemble-based constrained assimilation algorithms. These issues include localization in the presence of constraints, and the (possible) high dimensionality induced by the constraint systems. We use one 2D and one 3D case studies to demonstrate the performance of the proposed workflow. In particular, the 3D example contains experiment settings close to those of real field case studies. In both case studies, the proposed workflow achieves better data assimilation performance in comparison to the choice of using an original IES algorithm. As such, the proposed workflow has the potential to further improve the efficacy of ensemble-based data assimilation in practical reservoir data assimilation problems. PubDate: 2022-06-01
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Abstract: Abstract The accuracy and the limits of validity of the discontinuous pressure model, which describes fluid flow inside a fracture using a subgrid scale approach, is assessed by comparing simulation results with those from direct simulation using Stokes flow. While the subgrid scale approach assumes a unidirectional flow, the Stokes model includes both velocity components. This is at the cost of meshing the interior of the fracture, which is here achieved through a spline-based mesh generation scheme. This scheme explicitly couples the spline representing the discontinuity to the fracture mesh and thereby alleviates the (re)meshing requirements for the interior of the fracture. The subgrid model and the direct simulation of Stokes flow approaches are compared by simulating a typical case containing a pressurised fracture, highlighting the advantages of using a subgrid model for the range in which its assumptions are valid, and showing its capabilities to accurately include the influence of the fracture on the porous material even outside this range. PubDate: 2022-06-01
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Abstract: Abstract Thermal-hydro-mechanical (THM) coupled fracture propagation is common in underground engineering. Rock damage, as an inherent property of rock, significantly affects fracture propagation, but how it influences the THM coupled fracturing remains stubbornly unclear. A pore-scale THM coupling model is developed to study this problem, which combines the lattice Boltzmann method (LBM), the discrete element method (DEM), and rock damage development theory together for the first time. This model can more accurately calculate the exchanged THM information at the fluid-solid boundary and fluid conductivity dependent on fracture and rock damage. Based on the developed model, the synergistic effect of injected temperature difference (fluid temperature below rock temperature) and rock damage (characterized by the parameter “critical fracture energy”, abbreviated as “CFE”) on fracture propagation of shale are investigated particularly. It is found that: (1) the generation of branched cracks is closely related to the temperature response frontier, and the fracture process zone of single bond failure increases in higher CFE. (2) through the analysis of micro failure events, hydraulic fracturing is more pronounced in the low CFE, while thermal fracturing displays the opposite trend. The fluid conductivity of fractured rock increases with a higher injected temperature difference due to the more penetrated cracks and wider fracture aperture. However, this enhancement weakens when rock damage is significant. (3) in the multiple-layered rock with various CFEs, branched cracks propagating to adjacent layers are more difficult to form when the injection hole stays in the layer with significant rock damage than without rock damage. PubDate: 2022-05-16
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Abstract: Abstract Numerical simulation of surfactant flooding using conventional reservoir simulation models can lead to unreliable forecasts and bad decisions due to the appearance of numerical effects. The simulations give approximate solutions to systems of nonlinear partial differential equations describing the physical behavior of surfactant flooding by combining multiphase flow in porous media with surfactant transport. The approximations are made by discretization of time and space which can lead to spurious pulses or deviations in the model outcome. In this work, the black oil model was simulated using the decoupled implicit method for various conditions of reservoir scale models to investigate behavior in comparison with the analytical solution obtained from fractional flow theory. We investigated changes to cell size and time step as well as the properties of the surfactant and how it affects miscibility and flow. The main aim of this study was to understand pulse like behavior in the water bank, which we report for the first time, Our aim was to identify their cause and associated conditions. The pulses are induced by a sharp change in relative permeability as the interfacial tension changes. Pulses are diminished when adsorption is modeled, and ranged from 0.0002 kg/kg to 0.0005 kg/kg. The pulses are absent for high-resolution model of 5000 cells in x direction with a typical cell size as used in well-scale models. The growth or dampening of these pulses may vary from case to case but in this instance was a result of the combined impact of relative mobility, numerical dispersion, interfacial tension and miscibility. Oil recovery under the numerical problems reduced the performance of the flood, due to large amounts of pulses produced. Thus, it is important to improve existing models and use appropriate guidelines to stop oscillations and remove errors. PubDate: 2022-05-16
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Abstract: Abstract Artificial neural network-trained models were used to predict gas hydrate saturation distributions in permafrost-associated deposits in the Eileen Gas Hydrate Trend on the Alaska North Slope (ANS), USA and at the Mallik research site in the Beaufort-Mackenzie Basin, Northwest Territories, Canada. The database of Logging-While-Drilling (LWD) and wireline logs collected at five wells (Mount Elbert, Iġnik Sikumi, and Kuparuk 7–11–12 wells at ANS, plus 2L-38 and 5L-38 wells at the Mallik research site) includes more than 10,000 depth points, which were used for training, validation, and testing the machine learning (ML) models. Data used in training the ML models include the well logs of density, porosity, electrical resistivity, gamma radiation, and acoustic wave velocity measurements. Combinations of two or three out of these five well logs were found to reliably predict the gas hydrate saturation with accuracy varying between 80 and 90% when compared to the gas hydrate saturations derived from Nuclear Magnetic Resonance (NMR)-based technique. The ML models trained on data from three ANS wells achieved high fidelity predictions of gas hydrate saturation at the Mallik site. The results obtained in this study indicate that ML models trained on data from one geological basin can successfully predict key reservoir parameters for permafrost-associated gas hydrate accumulations within another basin. A generalized approach for selecting a well log combination that can improve model accuracy is discussed. Overall, the study outcome supports earlier work demonstrating that ML models trained on non-NMR well logs are a viable alternative to physics-driven methods for predicting gas hydrate saturations. PubDate: 2022-05-14
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Abstract: Abstract How to reconstruct a credible three-dimensional (3D) geological model from very limited survey data, e.g. boreholes, outcrop, and two-dimensional (2D) images, is challenging in the field of 3D geological modeling. Against the limitations of the huge computational consumption and complex parameterization of geostatistics-based stochastic simulation methods, we propose an automatic reconstruction method of 3D geological models based on deep convolutional generative adversarial network (DCGAN). In this work, 2D geological sections are used as conditioning data to generate 3D geological models automatically. Various realizations can be reproduced under a same DCGAN model established through deep network training. A U-Net structure is used to enhance the fitting effect of the DCGAN model. In addition, joint loss functions are exploited to increase the similarity between 3D realizations and reference models. Three synthetic datasets were used to verify the capability of the method presented in this paper. Experimental results show that the proposed 3D automatic reconstruction method based on DCGAN can capture the features, trends and spatial patterns of geological structures well. The output models obey the used conditioning data. The complex heterogeneous structures are reconstructed more accurately and quickly by using the proposed method. PubDate: 2022-05-10
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Abstract: Abstract It is well known that the construction of traditional reservoir simulation models can be very time and resources consuming. Particularly in the case of mature fields with long history and large number of wells where such models can be extremely difficult and long to history match. In this case data driven models can represent a cost-effective alternative, or they can provide complementary analysis to classical reservoir modelling. Due to data scarcity full machine learning approaches are also usually doomed to fail. In this work we develop a new Physics-Constrained Deep Learning approach that combined neural networks with a reduced physics approach: Capacitance Resistive Model (CRM). CRM are data-driven methods that are based on a simple material balance approximation, that can provide very useful reservoir insight. CRM can be used to analyze the underlying connections between producer wells and injector wells that can then be used to better allocate water injection. Such analysis can usually require very long tracer tests or very expensive 4D seismic acquisition and interpretation. CRM can provide directly these wells connection information using only available production and pressure data. The problem with CRM approaches, based on classical optimizers, is that they often detect spurious correlations and can be not very robust and reliable. Our physics-constrained deep learning approach called Deep-CRM performs production data regularization via the neural network approximation that helps to provide a better CRM parameter identification also with the use of robust gradient descent optimization methods developed and widely used by the large deep learning community. We show first on a synthetic and then in real reservoir case that Deep-CRM was able to identify most of the injector-producer connections with higher accuracy with respect to traditional CRM. Deep-CRM produced also better liquid production forecasts on the performed blind tests. PubDate: 2022-05-10
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Abstract: Abstract Damage in subsurface formations caused by mineral precipitation decreases the porosity and permeability, eventually reducing the production rate of wells in plants producing oil, gas or geothermal fluids. A possible solution to this problem consists in stopping the production followed by the injection of inhibiting species that slow down the precipitation process. In this work we model inhibitor injection and quantify the impact of a set of model parameters on the outputs of the system. The parameters investigated concern three key factors contributing to the success of the treatment: i) the inhibitor affinity, described by an adsorption Langmuir isotherm, ii) the concentration and time related to the injection and iii) the efficiency of the inhibitor in preventing mineral precipitation. Our simulations are set in a stochastic framework where these inputs are characterized in probabilistic terms. Forward simulations rely on a purpose-built code based on finite differences approximation of the reactive transport setup in radial coordinates. We explore the sensitivity diverse outputs, encompassing the well bottom pressure and space-time scales characterizing the transport of the inhibitor. We find that practically relevant output variables, such as inhibitor lifetime and well bottom pressure, display a diverse response to input uncertainties and display poor mutual dependence. Our results quantify the probability of treatment failure for diverse scenarios of inhibitor-rock affinity. We find that treatment optimization based on single outputs may lead to high failure probability when evaluated in a multi-objective framework. For instance, employing an inhibitor displaying an appropriate lifetime may fail in satisfying criteria set in terms of well-bottom pressure history or injected inhibitor mass. PubDate: 2022-04-30
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Abstract: Abstract Numerical models representing geological reservoirs can be used to forecast production and help engineers to design optimal development plans. These models should be as representative as possible of the true dynamic behavior and reproduce available static and dynamic data. However, identifying models constrained to production data can be very challenging and time consuming. Machine learning techniques can be considered to mimic and replace the fluid flow simulator in the process. However, the benefit of these approaches strongly depends on the simulation time required to train reliable predictors. Previous studies highlighted the potential of the multi-fidelity approach rooted in cokriging to efficiently provide accurate estimations of fluid flow simulator outputs. This technique consists in combining simulation results obtained on several levels of resolution for the reservoir model to predict the output properties on the finest level (the most accurate one). The degraded levels can correspond for instance to a coarser discretization in space or time, or to less complex physics. The idea behind is to take advantage of the coarse level low-cost information to limit the total simulation time required to train the meta-models. In this paper, we propose a new sequential design strategy for iteratively and automatically training (kriging and) cokriging based meta-models. As highlighted on two synthetic cases, this approach makes it possible to identify training sets leading to accurate estimations for the error between measured and simulated production data (objective function) while requiring limited simulation times. PubDate: 2022-04-26
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Abstract: Abstract In order to study the efficiency of the various forms of trapping including mineral trapping scenarios for CO2 storage behavior in deep layers of porous media, highly nonlinear coupled diffusion-advection-reaction partial differential equations (PDEs) including kinetic and equilibrium reactions modeling the miscible multiphase multicomponent flow have to be solved. We apply the globally fully implicit PDE reduction method (PRM) developed 2007 by Kräutle and Knabner for one-phase flow, which was extended 2019 to the case of two-phase flow with a pure gas in the study of Brunner and Knabner. We extend the method to the case of an arbitrary number of gases in gaseous phase, because CO2 is not the only gas that threats the climate, and usually is accompanied by other climate killing gases. The application of the PRM leads to an equation system consisting of PDEs, ordinary differential equations, and algebraic equations. The Finite Element discretized / Finite Volume stabilized equations are separated into a local and a global system but nevertheless coupled by the resolution function and evaluated with the aid of a nested Newton solver, so our solver is fully global implicit. For the phase disappearance, we use persistent variables which lead to a semismooth formulation that is solved with a semismooth Newton method. We present scenarios of the injection of a mixture of various gases into deep layers, we investigate phase change effects in the context of various gases, and study the mineral trapping effects of the storage technique. The technical framework also applies to other fields such as nuclear waste storage or oil recovery. PubDate: 2022-04-18
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Abstract: Abstract With the development of automatic measurement and data storage, vast quantities of data can be recorded and analyzed for heat transfer processes, which provides an opportunity to discover the transient heat transfer governing laws from the data. In this study, a machine learning-based sequential threshold ridge regression (STRidge) approach is applied to extract partial differential equations (PDEs) and tested on the heat conduction equation and conductive–convective heat transfer equation subjected to different boundary conditions, data volumes, and noise levels. Moreover, we studied the learning of governing equation of nonlinear transient heat transfer and used the improved STRidge with genetic algorithm to learn PDE with incomplete candidate library. The results showcase highly accurate identification of governing equations for heat transfer. And our results reveal the vast potential of the data-driven method in complex geothermal problems. PubDate: 2022-04-15
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Abstract: Abstract Estimation of porosity at a millimeter scale would be an order of magnitude finer resolution than traditional logging techniques. This enables proper description of reservoirs with thin layers and fine scale heterogeneities. To achieve this, we propose an end-to-end convolutional neural network (CNN) regression model that automatically predicts continuous porosity at a millimeter scale resolution using two-dimensional whole core CT scan images. More specifically, a CNN regression model is trained to learn from routine core analysis (RCA) porosity measurements. To characterize the performance of such approach, we compare the performance of this model with two linear regression models trained to learn the relationship between the average attenuation and standard deviation of the same two-dimensional images and RCA porosity. Our investigations reveal that the linear models are outperformed by the CNN, indicating the capability of the CNN model in extracting textures that are important for porosity estimations. We compare the predicted porosity results against the total porosity logs calculated from the density log. The obtained results show that the predicted porosity values using the proposed CNN method are well correlated with the core plug measurements and the porosity log. More importantly, the proposed approach can provide accurate millimeter scale porosity estimations, while the total porosity log is averaged over an interval and thus do not show such fine scale variations. Thus, the proposed method can be employed to calibrate the porosity logs, thereby reducing the uncertainties associated with indirect calculations of the porosity from such logs. PubDate: 2022-04-14
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Abstract: Abstract We investigate reactive flow and transport in evolving porous media. Solute species that are transported within the fluid phase are taking part in mineral precipitation and dissolution reactions for two competing mineral phases. The evolution of the three phases is not known a-priori but depends on the concentration of the dissolved solute species. To model the coupled behavior, phase-field and level-set models are formulated. These formulations are compared in three increasingly challenging setups including significant mineral overgrowth. Simulation outcomes are examined with respect to mineral volumes and surface areas as well as derived effective quantities such as diffusion and permeability tensors. In doing so, we extend the results of current benchmarks for mineral dissolution/precipitation at the pore-scale to the multiphasic solid case. Both approaches are found to be able to simulate the evolution of the three-phase system, but the phase-field model is influenced by curvature-driven motion. PubDate: 2022-04-14
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Abstract: Abstract Random reconstruction of three-dimensional (3D) digital rocks from two-dimensional (2D) slices is crucial for elucidating the microstructure of rocks and its effects on pore-scale flow in terms of numerical modeling, since massive samples are usually required to handle intrinsic uncertainties. Despite remarkable advances achieved by traditional process-based methods, statistical approaches and recently famous deep learning-based models, few works have focused on producing several kinds of rocks with one trained model and allowing the reconstructed samples to approximately satisfy certain given properties, such as porosity. To fill this gap, we propose a new framework with deep learning, named RockGPT, which is composed of VQ-VAE and conditional GPT, to synthesize 3D samples based on a single 2D slice from the perspective of video generation. The VQ-VAE is utilized to compress high-dimensional input video, i.e., the sequence of continuous rock slices, to discrete latent codes and reconstruct them. In order to obtain diverse reconstructions, the discrete latent codes are modeled using conditional GPT in an autoregressive manner, while incorporating conditional information from a given slice, rock type, and porosity. We conduct two experiments on five kinds of rocks, and the results demonstrate that RockGPT can produce different kinds of rocks with a single model, and the porosities of reconstructed samples can distribute around specified targets with a narrow range. In a broader sense, through leveraging the proposed conditioning scheme, RockGPT constitutes an effective way to build a general model to produce multiple kinds of rocks simultaneously that also satisfy user-defined properties. PubDate: 2022-04-11
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Abstract: Abstract This work investigates the performance of the on-demand machine learning (ODML) algorithm introduced in Leal et al. (Transp. Porous Media 133(2), 161–204, 2020) when applied to different reactive transport problems in heterogeneous porous media. This approach was devised to accelerate the computationally expensive geochemical reaction calculations in reactive transport simulations. We demonstrate that even with a strong heterogeneity present, the ODML algorithm speeds up these calculations by one to three orders of magnitude. Such acceleration, in turn, significantly advances the entire reactive transport simulation. The performed numerical experiments are enabled by the novel coupling of two open-source software packages: Reaktoro (Leal 2015) and Firedrake (Rathgeber et al. ACM Trans. Math. Softw. 43(3), 2016). The first library provides the most recent version of the ODML approach for the chemical equilibrium calculations, whereas, the second framework includes the newly implemented conservative Discontinuous Galerkin finite element scheme for the Darcy problem, i.e., the Stabilized Dual Hybrid Mixed (SDHM) method Núñez et al. (Int. J. Model. Simul. Petroleum Industry, 6, 2012). PubDate: 2022-04-01 DOI: 10.1007/s10596-021-10126-2
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