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- Correction to: Sequential fully implicit newton method for flow and
transport with natural black-oil formulation-
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PubDate: 2023-03-21
- A natural Hessian approximation for ensemble based optimization
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Abstract: A key challenge in reservoir management and other fields of engineering involves optimizing a nonlinear function iteratively. Due to the lack of available gradients in commercial reservoir simulators the attention over the last decades has been on gradient free methods or gradient approximations. In particular, the ensemble-based optimization has gained popularity over the last decade due to its simplicity and efficient implementation when considering an ensemble of reservoir models. Typically, a regression type gradient approximation is used in a backtracking or line search setting. This paper introduces an approximation of the Hessian utilizing a Monte Carlo approximation of the natural gradient with respect to the covariance matrix. This Hessian approximation can further be implemented in a trust region approach in order to improve the efficiency of the algorithm. The advantages of using such approximations are demonstrated by testing the proposed algorithm on the Rosenbrock function and on a synthetic reservoir field.
PubDate: 2023-03-16
- Three-dimensional numerical simulation of hydraulically driven cohesive
fracture propagation in deformable reservoir rock using enriched EFG
method-
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Abstract: In this paper, a fully coupled 3D. numerical simulation of hydraulic fracture propagation in saturated deformable porous media is presented in the context of the extrinsically enriched element free Galerkin (EFG) method. By exploiting the partition of unity property of moving least square shape functions, weak and strong discontinuities are simulated using the Ridge and the Heaviside enrichment functions, respectively. The cohesive crack model is used to describe the nonlinear fracture processes developing in the area in front of the crack tip where the energy dissipation takes place. The fracturing fluid flow within the fracture is modeled using Darcy’s law and the fracture permeability is considered to follow the cubic law. The developed fully coupled numerical framework can simulate the fluid leak-off phenomenon and formation of the fluid-lag zone. For verification of the developed computational algorithm, a problem with an analytical solution was simulated and a good agreement was seen between numerical and analytical results. The numerical simulations and the parametric studies results show that the proposed numerical framework can successfully simulate various aspects of the complicated process of the hydraulic fracturing treatment.
PubDate: 2023-03-15
- Comparison of nonlinear field-split preconditioners for two-phase flow in
heterogeneous porous media-
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Abstract: This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approach—and in particular, FSMSN—results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance–i.e., measured by wall-clock time–will be studied in a subsequent publication.
PubDate: 2023-03-15
- A posteriori error analysis of a Banach spaces-based fully mixed FEM for
double-diffusive convection in a fluid-saturated porous medium-
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Abstract: In this paper we consider a Banach spaces-based fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, and due to the nonlinear nature of the problem, we employ the strong monotonicity of the operator involving the Forchheimer term, in addition to inf-sup conditions of some of the resulting bilinear forms, along with a stable Helmholtz decomposition in nonstandard Banach spaces, which, in turn, having been recently derived, constitutes another distinctive feature of the paper, and local approximation properties of the Raviart–Thomas and Clément interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with an irregular channel networks is considered.
PubDate: 2023-03-08
- Sequential multilevel assimilation of inverted seismic data
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Abstract: We consider estimation of absolute permeability from inverted seismic data. Large amounts of simultaneous data, such as inverted seismic data, enhance the negative effects of Monte Carlo errors in ensemble-based Data Assimilation (DA). Multilevel (ML) models consist of a selection of models with different fidelities. Multilevel Data Assimilation (MLDA) attempts to obtain a better statistical accuracy with a small sacrifice of the numerical accuracy. Spatial grid coarsening is one way of generating an ML model. It has been shown that coarsening the spatial grid results in a problem with weaker nonlinearity, and hence, in a less challenging problem than the problem on the original fine grid. Accordingly, formulating a sequential MLDA algorithm which uses the coarser models in the first steps of the DA, followed by the finer models, helps to find an approximation to the solution of the inverse problem at the first steps and gradually converge to the solution. We present two variants of a sequential MLDA algorithm and compare their performance with both conventional DA algorithms and a simultaneous (i.e., using all the models on the different grids simultaneously) MLDA algorithm using numerical experiments. Both posterior parameters and posterior model forecasts are compared qualitatively and quantitatively. The results from numerical experiments suggest that all MLDA algorithms generally perform better than the conventional DA algorithms. In estimation of the posterior parameter fields, the simultaneous MLDA algorithm and one of the variants of sequential MLDA (SMLES-H) perform similarly and slightly better than the other variant (SMLES-S). While in estimation of the posterior model forecasts, SMLES-S clearly performs better than both the simultaneous MLDA algorithm and SMLES-H.
PubDate: 2023-02-18
- Acknowledgement for Reviewers for 2022
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PubDate: 2023-02-16
- The method of forced probabilities: a computation trick for Bayesian model
evidence-
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Abstract: Bayesian model selection objectively ranks competing models by computing Bayesian Model Evidence (BME) against test data. BME is the likelihood of data to occur under each model, averaged over uncertain parameters. Computing BME can be problematic: exact analytical solutions require strong assumptions; mathematical approximations (information criteria) are often strongly biased; assumption-free numerical methods (like Monte Carlo) are computationally impossible if the data set is large, for example like high-resolution snapshots from experimental movies. To use BME as ranking criterion in such cases, we develop the “Method of Forced Probabilities (MFP)”. MFP swaps the direction of evaluation: instead of comparing thousands of model runs on random model realizations with the observed movie snapshots, we force models to reproduce the data in each time step and record the individual probabilities of the model following these exact transitions. MFP is fast and accurate for models that fulfil the Markov property in time, paired with high-quality data sets that resolve all individual events. We demonstrate our approach on stochastic macro-invasion percolation models that simulate gas migration in porous media, and list additional examples of probable applications. The corresponding experimental movie was obtained from slow gas injection into water-saturated, homogeneous sand in a 25 x 25 x 1 cm acrylic glass tank. Despite the movie not always satisfying the high demands (resolving all individual events), we can apply MFP by suggesting a few workarounds. Results confirm that the proposed method can compute BME in previously unfeasible scenarios, facilitating a ranking among competing model versions for future model improvement.
PubDate: 2023-02-01
- Impact of geostatistical nonstationarity on convolutional neural network
predictions-
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Abstract: Convolutional neural networks (CNNs) are gaining tremendous attention in subsurface studies due to their ability to learn from spatial image data. However, most deep learning studies in spatial context do not consider the impact of geostatistical nonstationarity, which is commonly encountered within the subsurface phenomenon. We demonstrate the impact of geostatistical nonstationarity on CNN prediction performance. We propose a CNN model to predict the variogram range of sequential Gaussian simulation (SGS) realizations. Model performance is evaluated for stationarity and three common forms of geostatistical nonstationarity: (1) large relative variogram range-related nonstationarity, (2) additive trend and residual model-related nonstationarity, and (3) mixture population model-related nonstationarity. Our CNN model prediction accuracy decreases in the presence of large relative variogram range-related nonstationarity, for the additive trend and residual model-related nonstationarity, the relative prediction errors increase for high trend variance proportions with a decrease in variogram range; regarding the mixture population model-related nonstationarity, the predictions are closer to the smaller variogram range. Common forms of geostatistical nonstationarity may impact CNN predictions, as with geostatistical estimation methods, trend removal and workflows with stationary residuals are recommended.
PubDate: 2023-02-01
- Prediction of permeability of porous media using optimized convolutional
neural networks-
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Abstract: Permeability is an important parameter to describe the behavior of a fluid flow in porous media. To perform realistic flow simulations, it is essential that the fine scale models include permeability variability. However, these models cannot be used directly in simulations because require high computational cost, which motivates the application of upscaling approaches. In this context, machine learning techniques can be used as an alternative to perform the upscaling of porous media properties with lower computational cost than traditional upscaling methods. Hence, in this work, an upscaling methodology is proposed to compute the equivalent permeability on the large grid through convolutional neural networks (CNN). This method achieves suitable precision, with less computational demand, when evaluated on 2D and 3D models, if compared with the local upscaling approach. We also present a genetic algorithm (GA) to automatically determine the optimal configuration of CNNs for the target problems. The GA procedure is applied to yield the optimal CNN architecture for upscaling of the permeability fields with outstanding results when compared with counterpart techniques.
PubDate: 2023-02-01
- Estimating permeability of 3D micro-CT images by physics-informed CNNs
based on DNS-
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Abstract: In recent years, convolutional neural networks (CNNs) have experienced an increasing interest in their ability to perform a fast approximation of effective hydrodynamic parameters in porous media research and applications. This paper presents a novel methodology for permeability prediction from micro-CT scans of geological rock samples. The training data set for CNNs dedicated to permeability prediction consists of permeability labels that are typically generated by classical lattice Boltzmann methods (LBM) that simulate the flow through the pore space of the segmented image data. We instead perform direct numerical simulation (DNS) by solving the stationary Stokes equation in an efficient and distributed-parallel manner. As such, we circumvent the convergence issues of LBM that frequently are observed on complex pore geometries, and therefore, improve the generality and accuracy of our training data set. Using the DNS-computed permeabilities, a physics-informed CNN (PhyCNN) is trained by additionally providing a tailored characteristic quantity of the pore space. More precisely, by exploiting the connection to flow problems on a graph representation of the pore space, additional information about confined structures is provided to the network in terms of the maximum flow value, which is the key innovative component of our workflow. The robustness of this approach is reflected by very high prediction accuracy, which is observed for a variety of sandstone samples from archetypal rock formations.
PubDate: 2023-01-31
- Convolutional – recurrent neural network proxy for robust optimization
and closed-loop reservoir management-
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Abstract: Production optimization under geological uncertainty is computationally expensive, as a large number of well control schedules must be evaluated over multiple geological realizations. In this work, a convolutional-recurrent neural network (CNN-RNN) proxy model is developed to predict well-by-well oil and water rates, for given time-varying well bottom-hole pressure (BHP) schedules, for each realization in an ensemble. This capability enables the estimation of the objective function and nonlinear constraint values required for robust optimization. The proxy model represents an extension of a recently developed long short-term memory (LSTM) RNN proxy designed to predict well rates for a single geomodel. A CNN is introduced here to processes permeability realizations, and this provides the initial states for the RNN. The CNN-RNN proxy is trained using simulation results for 300 different sets of BHP schedules and permeability realizations. We demonstrate proxy accuracy for oil-water flow through multiple realizations of 3D multi-Gaussian permeability models. The proxy is then incorporated into a closed-loop reservoir management (CLRM) workflow, where it is used with particle swarm optimization and a filter-based method for nonlinear constraint satisfaction. History matching is achieved using an adjoint-gradient-based procedure. The proxy model is shown to perform well in this setting for five different (synthetic) ‘true’ models. Improved net present value along with constraint satisfaction and uncertainty reduction are observed with CLRM. For the robust production optimization steps, the proxy provides O(100) runtime speedup over simulation-based optimization.
PubDate: 2023-01-27
- Fast prediction of aquifer thermal energy storage: a multicyclic
metamodelling procedure-
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Abstract: The metamodel-based approach (also referred to as the surrogate approach) is commonly applied to overcome the computational burden of numerical models that are used to simulate the evolution of reservoir fluids and pressures in response to any production scheme. In this study, we propose an adaptation of this approach for aquifer thermal energy storage (ATES) systems. ATES systems are characterized by cyclic loading/unloading production schemes, which result in a strong similarity in the dynamics of the intercyclic evolution of variables such as the temperature at the producer well. Instead of training several metamodels, i.e., one per cycle (“independent” metamodelling approach), we take advantage of the intercyclic similarity to train a single metamodel within the setting of multifidelity cokriging (“multicyclic” metamodelling approach). To explore the predictive performance of this approach, we applied a random subsampling validation approach multiple times to 300 simulation results of a realistic ATES system in the Paris basin by considering three characteristics, i.e., the minimum and maximum temperature, and the rate of temperature decrease at each cycle. Numerical experiments with varying training dataset sizes (from 33 to 66% of the total number of results) and using 100 test samples show that (1) the predictive error of the multicyclic metamodelling reaches lower levels (by 20–50%) than that of the independent approach; (2) this higher predictive performance is achieved while saving computational time cost because the training phase only needs a few tens of “complete” simulations (run over all cycles) together with a few hundreds of “partial” simulations (stopped at the first cycle); the latter simulations are less expensive to evaluate because of shorter simulated time.
PubDate: 2023-01-27
- Geological realism in Fluvial facies modelling with GAN under variable
depositional conditions-
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Abstract: This study investigates generative adversarial networks (GANs)’ capacity to model multi-facies distributions of meandering systems. Earlier works showed that GANs outperform geostatistical methods in reproducing complex geometry, like the shapes of fluvial channels. However, the reproduction of geological complexity and geological realism remains an issue when modelling fluvial depositional systems. Meandering systems deposit multiple facies and change facies shape following the migration of rivers. Sand accretes at the inner bank of channels, forming the point bar and erodes the plain at the outer bank to create sediments. Channel fills with mud or sand at the bottom after abandonment due to avulsions or meander cut-offs. Those sedimentary processes yield complex geological patterns. This paper proposes further developing a GAN model, Fluvial GAN, to learn complex multi-facies fluvial patterns across depositional variability. We create a set of meandering facies models by a process-based model, FLUMYTM, for training a GAN and assessing how well it can learn fluvial facies distributions representing sedimentary processes. Fluvial GAN has three distinct enhancements: (i) a One-Hot Encoder for better handling of multi-facies distribution, (ii) a Hybrid-discriminator for better learning geological patterns, and (iii) an improved loss function to prevent mode collapse. We compare Fluvial GAN performance with two more standard configurations using qualitative and quantitative geological features assessments. Fluvial GAN vastly reduces the occurrence of a typical unrealistic feature, channels forming isolated loops, which we called ‘closed channel’ in this study. We analyse the diversity of Fluvial GAN generations via a dimensionality reduction algorithm, UMAP, that plots the training dataset and Fluvial GAN generations together in a 2D space. Fluvial GAN provides good coverage of the uncertainty space represented by the training dataset.
PubDate: 2023-01-27
- Dynamic time warping for well injection and production history
connectivity characterization-
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Abstract: Waterflooding is the most widely used secondary method for oil recovery. Therefore, determining well interference between injector and oil producer wells is crucial to achieving higher recovery factors. Several techniques are used to evaluate the connectivity between injectors and producer wells, such as correlation coefficients, linear regression models, and capacitance resistance models. However, such methods rely on limited flow physics and are based on various assumptions about the data, and subsurface, geological, and engineering settings. We propose a novel, intuitive method, with physics-constrained dynamic time warping algorithm (PCDTW) to detect the pairwise influence of water injection wells on oil production wells response by mapping the input water injection signal on the output oil production signal. The proposed signal mapping PCDTW method can efficiently determine the lag time between water injection and oil production response, to characterize the reservoir formation connectivity and heterogeneity between paired injection and production wells. Our proposed method is based on an enhanced physics-based model with constraints for subsurface flow through porous media to improve accuracy, avoid incorrect signal matches or non-physical results, and reduce uncertainty. Additionally, there are limited assumptions about the data and the method is robust in the presence of data measurement noise. Our proposed method is a data-driven, domain and physics informed model to support subsurface engineering data science.
PubDate: 2022-12-26
DOI: 10.1007/s10596-022-10188-w
- Analysis of earthquake hypocenter characteristics using chaos game
representation-
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Abstract: This paper proposes a new method to approach earthquake hypocenter studies based on Chaos Game Representation (CGR), a method initially used for making fractal structures and applied for studying DNA sequences. Applying the CGR method, this study aims at checking whether any relation exists between earthquakes occurring in different depth ranges in a seismically active area. For this purpose, the seismically active areas around the Indian tectonic plate were used. The CGR images gave characteristic patterns, implying that the occurrence of earthquakes in some specific depth range combinations showed higher preference. Statistical data on the frequency of different depth range combinations were derived from these plots. We put forward a mathematical value which we call proximity index, to compare the similarity between two different CGR plots. Proximity index values were used to compare the similarity in seismic activity in two different zones by comparing their respective CGR plots.
PubDate: 2022-12-26
DOI: 10.1007/s10596-022-10187-x
- Multi-level emulation of tsunami simulations over Cilacap, South Java,
Indonesia-
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Abstract: Carrying out a Probabilistic Tsunami Hazard Assessment (PTHA) requires a large number of simulations done at a high resolution. Statistical emulation builds a surrogate to replace the simulator and thus reduces computational costs when propagating uncertainties from the earthquake sources to the tsunami inundations. To reduce further these costs, we propose here to build emulators that exploit multiple levels of resolution and a sequential design of computer experiments. By running a few tsunami simulations at high resolution and many more simulations at lower resolutions we are able to provide realistic assessments whereas, for the same budget, using only the high resolution tsunami simulations do not provide a satisfactory outcome. As a result, PTHA can be considered with higher precision using the highest spatial resolutions, and for impacts over larger regions. We provide an illustration to the city of Cilacap in Indonesia that demonstrates the benefit of our approach.
PubDate: 2022-12-21
DOI: 10.1007/s10596-022-10183-1
- Algebraic flux correction finite element method with semi-implicit time
stepping for solute transport in fractured porous media-
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Abstract: This work is concerned with the numerical modeling of the Darcy flow and solute transport in fractured porous media for which the fractures are modeled as interfaces of codimension one. The hybrid-dimensional flow and transport problems are discretizaed by a lumped piece-wise linear finite element method, combined with the algebraic correction of the convective fluxes. The resulting transport discretization can be interpreted as a conservative finite volume scheme that satisfies the discrete maximum principle, while introducing a very limited amount of numerical diffusion. In the context of fractured porous media flow the CFL number may vary by several order of magnitude, which makes explicit time stepping unfeasible. To cope with this difficulty we propose an adaptive semi-implicit time stepping strategy that reduces to the low order linear implicit discretization in the high CFL regions that include, but may not be limited to the fracture network. The performance of the fully explicit and semi-implicit variants of the method are investigated through the numerical experiment.
PubDate: 2022-12-13
DOI: 10.1007/s10596-022-10178-y
- Iterative solution methods for 3D controlled-source electromagnetic
forward modelling of geophysical exploration scenarios-
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Abstract: We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-valued system and solve this arising real-valued two-by-two block system (outer system) using the generalised conjugate residual method preconditioned with a highly efficient block-based PREconditioner for Square Blocks (PRESB). Applying this preconditioner equates to solving two smaller inner symmetric systems which are either solved using a direct solver or iterative methods, namely the generalised conjugate residual or the flexible generalised minimal residual methods preconditioned with the multigrid-based auxiliary-space preconditioner AMS. Our numerical experiments demonstrate the robustness of the outer solver with respect to spatially variable material parameters, for a wide frequency range of five orders of magnitude (0.1-10’000 Hz), with respect to the number of degrees of freedom, and for stretched structured and unstructured as well as locally refined meshes. For all the models considered, the outer solver reaches convergence in a small (typically < 20) number of iterations. Further, our numerical tests clearly show that solving the two inner systems iteratively using the indicated preconditioned iterative methods is computationally beneficial in terms of memory requirement and time spent as compared to a direct solver. On top of that, our iterative framework works for large-scale problems where direct solvers applied to the original complex-valued systems succumb due to their excessive memory consumption, thus making the iterative framework better suited for large-scale 3D problems. Comparison to a similar iterative framework based on a block-diagonal and the auxiliary-space preconditioners reveals that the PRESB preconditioner requires slightly fewer iterations to converge yielding a certain gain in time spent to obtain the solution of the two-by-two block system.
PubDate: 2022-12-06
DOI: 10.1007/s10596-022-10182-2
- Continuous and discrete data assimilation with noisy observations for the
Rayleigh-Bénard convection: a computational study-
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Abstract: Obtaining accurate high-resolution representations of model outputs is essential to describe the system dynamics. In general, however, only spatially- and temporally-coarse observations of the system states are available. These observations can also be corrupted by noise. Downscaling is a process/scheme in which one uses coarse scale observations to reconstruct the high-resolution solution of the system states. Continuous Data Assimilation (CDA) is a recently introduced downscaling algorithm that constructs an increasingly accurate representation of the system states by continuously nudging the large scales using the coarse observations. We introduce a Discrete Data Assimilation (DDA) algorithm as a downscaling algorithm based on CDA with discrete-in-time nudging. We then investigate the performance of the CDA and DDA algorithms for downscaling noisy observations of the Rayleigh-Bénard convection system in the chaotic regime. In this computational study, a set of noisy observations was generated by perturbing a reference solution with Gaussian noise before downscaling them. The downscaled fields are then assessed using various error- and ensemble-based skill scores. The CDA solution was shown to converge towards the reference solution faster than that of DDA but at the cost of a higher asymptotic error. The numerical results also suggest a quadratic relationship between the ℓ2 error and the noise level for both CDA and DDA. Cubic and quadratic dependences of the DDA and CDA expected errors on the spatial resolution of the observations were obtained, respectively.
PubDate: 2022-12-05
DOI: 10.1007/s10596-022-10180-4
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