We present a novel physics-informed deep learning framework for solving steady-state incompressible flow on multiple sets of irregular geometries by incorporating two main elements: using a ...point-cloud based neural network to capture geometric features of computational domains, and using the mean squared residuals of the governing partial differential equations, boundary conditions, and sparse observations as the loss function of the network to capture the physics. While the solution of the continuity and Navier-Stokes equations is a function of the geometry of the computational domain, current versions of physics-informed neural networks have no mechanism to express this functionally in their outputs, and thus are restricted to obtain the solutions only for one computational domain with each training procedure. Using the proposed framework, three new facilities become available. First, the governing equations are solvable on a set of computational domains containing irregular geometries with high variations with respect to each other but requiring training only once. Second, after training the introduced framework on the set, it is now able to predict the solutions on domains with unseen geometries from seen and unseen categories as well. The former and the latter both lead to savings in computational costs. Finally, all the advantages of the point-cloud based neural network for irregular geometries, already used for supervised learning, are transferred to the proposed physics-informed framework. The effectiveness of our framework is shown through the method of manufactured solutions and thermally-driven flow for forward and inverse problems.
•PIPN is a novel physics-informed deep learning framework.•PIPN solves forward and inverse time-independent problems on several irregular domains by training only once.•PIPN overcomes the shortcoming of regular PINNs that need to be retrained for any single domain with a new geometry.•Applications of PIPN are shown for incompressible flows and thermal fields.
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Quantitative Seismic Interpretation demonstrates how rock physics can be applied to predict reservoir parameters, such as lithologies and pore fluids, from seismically derived attributes. The authors ...provide an integrated methodology and practical tools for quantitative interpretation, uncertainty assessment, and characterization of subsurface reservoirs using well-log and seismic data. They illustrate the advantages of these new methodologies, while providing advice about limitations of the methods and traditional pitfalls. This book is aimed at graduate students, academics and industry professionals working in the areas of petroleum geoscience and exploration seismology. It will also interest environmental geophysicists seeking a quantitative subsurface characterization from shallow seismic data. The book includes problem sets and a case-study, for which seismic and well-log data, and Matlab codes are provided on a website (http://www.cambridge.org/9780521816014). These resources will allow readers to gain a hands-on understanding of the methodologies.
Geological facies modeling has long been studied to predict subsurface resources. In recent years, generative adversarial networks (GANs) have been used as a new method for geological facies modeling ...with surprisingly good results. However, in conventional GANs, all layers are trained concurrently, and the scales of the geological features are not considered. In this study, we propose to train GANs for facies modeling based on a new training process, namely progressive growing of GANs or a progressive training process. In the progressive training process, GANs are trained layer by layer, and geological features are learned from coarse scales to fine scales. We also train a GAN in the conventional training process, and compare the conventionally trained generator with the progressively trained generator based on visual inspection, multi-scale sliced Wasserstein distance (MS-SWD), multi-dimensional scaling (MDS) plot visualization, facies proportion, variogram, and channel sinuosity, width, and length metrics. The MS-SWD reveals realism and diversity of the generated facies models, and is combined with MDS to visualize the relationship between the distributions of the generated and training facies models. The conventionally and progressively trained generators both have very good performances on all metrics. The progressively trained generator behaves especially better than the conventionally trained generator on the MS-SWD, MDS plots, and the necessary training time. The training time for the progressively trained generator can be as small as 39% of that for the conventionally trained generator. This study demonstrates the superiority of the progressive training process over the conventional one in geological facies modeling, and provides a better option for future GAN-related researches.
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Cave geometry, spatial distribution, and interconnectivity are critical for developing resource production and contaminant remediation strategies. Geologically realistic stochastic models for ...simulating karst are essential for quantifying the spatial uncertainty of karst networks given geophysical observations. Dynamic graph dissolution, a novel physics-based approach for three-dimensional stochastic geomodeling of telogenetic karst morphology, is introduced herein. The cave evolution is modeled through dissolution of fractures over geologic time based on a graph representation of discrete fracture networks, which can be informed by field observations. The graph is initially modeled based on fracture intersections. In order to account for overlapping enlargements, the graph representation is updated over dissolution using the Mapper algorithm with density-based spatial clustering. This modeling approach enables the generation of multiple realizations of different geologic scenarios of karst formation at a tractable computational cost. Realizations generated using the proposed algorithm are compared with real caves using graph topological metrics such as central point dominance, connectivity degree, average degree, degree dispersion, and assortativity. The distributions of graph topological metrics of generated realizations overlap with the metrics of known caves, suggesting that the graph structures of observed and simulated caves are at least globally similar.
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We present a framework that enables estimation of low-dimensional sub-resolution reservoir properties directly from seismic data, without requiring the solution of a high dimensional seismic inverse ...problem. Our workflow is based on the Bayesian evidential learning approach and exploits learning the direct relation between seismic data and reservoir properties to efficiently estimate reservoir properties. The theoretical framework we develop allows incorporation of non-linear statistical models for seismic estimation problems. Uncertainty quantification is performed with approximate Bayesian computation. With the help of a synthetic example of estimation of reservoir net-to-gross and average fluid saturations in sub-resolution thin sand reservoir, several nuances are foregrounded regarding the applicability of unsupervised and supervised learning methods for seismic estimation problems. Finally, we demonstrate the efficacy of our approach by estimating posterior uncertainty of reservoir net-to-gross in sub-resolution thin sand reservoir from an offshore delta dataset using 3D pre-stack seismic data.
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Conditional facies modeling combines geological spatial patterns with different types of observed data, to build earth models for predictions of subsurface resources. Recently, researchers have used ...generative adversarial networks (GANs) for conditional facies modeling, where an unconditional GAN is first trained to learn the geological patterns using the original GAN’s loss function, then appropriate latent vectors are searched to generate facies models that are consistent with the observed conditioning data. A problem with this approach is that the time-consuming search process needs to be conducted for every new conditioning data. As an alternative, we improve GANs for conditional facies simulation (called GANSim) by introducing an extra condition-based loss function and adjusting the architecture of the generator to take the conditioning data as inputs, based on progressive growing of GANs. The condition-based loss function is defined as the inconsistency between the input conditioning value and the corresponding characteristics exhibited by the output facies model, and forces the generator to learn the ability of being consistent with the input conditioning data, together with the learning of geological patterns. Our input conditioning factors include global features (e.g., the mud facies proportion) alone, local features such as sparse well facies data alone, and joint combination of global features and well facies data. After training, we evaluate both the quality of generated facies models and the conditioning ability of the generators, by manual inspection and quantitative assessment. The trained generators are quite robust in generating high-quality facies models conditioned to various types of input conditioning information.
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Data assimilation methods are commonly used to predict petrophysical properties of deep saline aquifers for carbon dioxide sequestration studies. However, data assimilation is usually computationally ...challenging for large-scale geological models with a large number of geophysical observations. A computationally efficient method is developed for large-scale data assimilation problems. In the proposed method, model parameters and observations are reduced by randomized tensor decomposition, and the inversion is performed in lower-dimensional spaces. Tensors are multiway arrays that generalize matrices to multiple dimensions and therefore provide a natural way to represent three-dimensional geological models and geophysical measurements. To exploit the inherent multidimensional structure of tensors, high-order singular value decomposition performs orthogonal decomposition of the tensor in the high-order space instead of using a flattened matrix representation. However, for complex geological models, the dimension of tensors becomes very large, making the decomposition computationally unfeasible. To overcome this limitation, the dimension of the tensors is first reduced by a randomized linear algebra algorithm which guarantees that most of the information of the original tensors is preserved with high probability in the reduced tensor, and then high-order singular value decomposition is applied on the smaller tensors with relatively small computational cost. By using randomized tensor decomposition for model and data re-parameterization, data assimilation is efficiently performed in the low-dimensional model and data space. The proposed method is demonstrated and validated on the three-dimensional data set synthesized based on the Johansen formation, a potential carbon dioxide storage site offshore in Norway.
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Geological facies modeling is a key component in exploration and characterization of subsurface reservoirs. While traditional geostatistical approaches are still commonly used nowadays, deep learning ...is gaining a lot of attention within geoscientific community for generating subsurface models, as a result of recent advance of computing powers and increasing availability of training data sets. This work presents a deep learning approach for geological facies modeling based on generative adversarial networks (GANs) combined with training-image-based simulation. In a typical application of learned networks, all neural network parameters are fixed after training, and the uncertainty in the trained model cannot be analyzed. To address this problem, a Bayesian GANs (BGANs) approach is proposed to create facies models. In this approach, a probability distribution is assigned to the neural parameters of the BGANs. Only neural parameters of the generator in BGANs are assigned with a probability function, and the ones in the discriminator are treated as fixed. Random samples are then drawn from the posterior distribution of neural parameters to simulate subsurface facies models. The proposed approach is applied to the two different geological depositional scenarios, fluvial channels and carbonate mounds, and the generated models reasonably capture the variability of the training/testing data. Meanwhile, the model uncertainty of learned networks is readily accessible. To fully sample the spatial distribution in the training image set, a large collection of samples of network parameters is required to be drawn from the posterior distribution, thus significantly increasing computational cost.
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The knowledge of fracture properties and its geometrical patterns is often required for the analysis of mechanical and flow properties in fractured reservoirs, as fracture characterization plays a ...critical role in the optimization of hydrocarbon production or estimation of storage capacity of subsurface reservoirs. A stochastic method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to estimate fracture properties using a rock physics model for fractured rocks. Two implementations are presented: a Metropolis algorithm based on a Gaussian prior distribution and an extended Metropolis algorithm with an informative prior obtained from multiple-point statistics simulations. The results are compared to a Bayesian analytical approach where the solution is based on a linearized approximation of the rock physics model. The novelty of the proposed approach is the use of a training image, that is, a conceptual geological model, to account for the spatial distribution of the fractures. Two fracture properties are considered, namely fracture density and aspect ratio, and the spatial distribution and geometrical characteristics of fractures are also investigated to understand the connectivity patterns that control fluid flow. The MCMC approach with a training image is more computationally demanding but provides geometrical models of the spatial distribution of fractures. The inversion results show that the prediction accuracy of fracture density and aspect ratio obtained by the MCMC methods is similar to the one obtained with the analytical approach, and that the MCMC methods provide a reliable assessment of the posterior uncertainty as well.
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