Mathematical Geosciences最新文献

Rule-based reservoir models incorporate rules that mimic actual sediment deposition processes for accurate representation of geological patterns of sediment accumulation. Bayesian methods combine rule-based reservoir modelling and well data, with geometry and placement rules as part of the prior and well data accounted for by the likelihood. The focus here is on a shallow marine shoreface geometry of ordered sedimentary packages called bedsets. Shoreline advance and sediment build-up are described through progradation and aggradation parameters linked to individual bedset objects. Conditioning on data from non-vertical wells is studied. The emphasis is on the role of ‘configurations’—the order and arrangement of bedsets as observed within well intersections in establishing the coupling between well observations and modelled objects. A conditioning algorithm is presented that explicitly integrates uncertainty about configurations for observed intersections between the well and the bedset surfaces. As data volumes increase and model complexity grows, the proposed conditioning method eventually becomes computationally infeasible. It has significant potential, however, to support the development of more complex models and conditioning methods by serving as a reference for consistency in conditioning.

Prediction of flow to boreholes or excavations in fractured low-permeability rocks is important for resource extraction and disposal or sequestration activities. Analytical solutions for fluid pressure and flowrate, when available, are powerful, insightful, and efficient tools enabling parameter estimation and uncertainty quantification. A flexible porous media flow solution for arbitrary physical dimensions is derived and extended to double porosity for converging radial flow when permeability and porosity decrease radially as a power law away from a borehole or opening. This distribution can arise from damage accumulation due to stress relief associated with drilling or mining. The single-porosity graded conductivity solution was initially found for heat conduction, the arbitrary dimension flow solution comes from hydrology, and the solution with both arbitrary dimension and graded permeability distribution appeared in reservoir engineering. These existing solutions are combined and extended here to two implementations of the double-porosity conceptual model, for both a simpler thin-film mass transfer and more physically realistic diffusion between fracture and matrix. This work presents a new specified-flowrate solution with wellbore storage for the simpler double-porosity model, and a new, more physically realistic solution for any wellbore boundary condition. A new closed-form expression is derived for the matrix diffusion solution (applicable to both homogeneous and graded problems), improving on previous infinite series expressions.

Distributional data have recently become increasingly important for understanding processes in the geosciences, thanks to the establishment of cost-efficient analytical instruments capable of measuring properties over large numbers of particles, grains or crystals in a sample. Functional data analysis allows the direct application of multivariate methods, such as principal component analysis, to such distributions. However, these are often observed in the form of samples, and thus incur a sampling error. This additional sampling error changes the properties of the multivariate variance and thus the number of relevant principal components and their direction. The result of the principal component analysis becomes an artifact of the sampling error and can negatively affect the subsequent data analysis. This work presents a way of estimating this sampling error and how to confront it in the context of principal component analysis, where the principal components are obtained as a linear combination of elements of a newly constructed orthogonal spline basis. The effect of the sampling error and the effectiveness of the correction is demonstrated with a series of simulations. It is shown how the interpretability and reproducibility of the principal components improve and become independent of the selection of the basis. The proposed method is then applied on a dataset of grain size distributions in a geometallurgical dataset from Thaba mine in the Bushveld complex.

Various deep learning algorithms (DLAs) have been successfully employed for mineral prospectivity mapping (MPM) to support mineral exploration, due to their superior nonlinear extraction capabilities. DLAs algorithms are typically purely data-driven approaches that may ignore the geological domain knowledge. This renders the predictive results inconsistent with the mineralization mechanism and results in poor interpretation. In this study, a geologically constrained convolutional neural network (CNN) that involves soft and hard geological constraints was proposed for mapping gold polymetallic mineralization potential in western Henan Province of China. A penalty term based on the controlling equation of the spatial coupling relationship between the ore-controlling strata and gold deposits was constructed as a soft constraint to guide the CNN model training according to additional prior geological knowledge. In addition, domain knowledge related to mineralization processes and a geochemical indicator were simultaneously embedded as hard constraints in the feature extractor and classifier of the CNN, respectively, to control the model training based on the mineralization mechanism. The comparative experiments demonstrated that the geologically constrained CNN was superior to other models, thus indicating that the coupling of data and domain knowledge is effective for MPM and further improves the rationality and interpretability of the obtained results.

Chemical analyses of powdered rocks by different laboratories often yield varying results, requiring estimation of the rock’s true composition and associated uncertainty. Challenges arise from the peculiar nature of geochemical data. Traditionally, major and trace elements have been measured using different methods, resulting in chemical analyses where the sum of the parts fluctuates around 1 rather than precisely totaling 1. Additionally, all chemical analyses contain an undisclosed mass fraction representing undetected chemical elements. Because of this undisclosed and unknown mass fraction, geochemical data represent a particular kind of compositional data in which closure to unity is not guaranteed. We argue that chemical analyses exist in the hypercube while being sampled from a true composition residing in the simplex. Therefore, we propose an algorithm that generates random chemical analyses by simulating the data acquisition protocol in geochemistry. Using the algorithm’s output, we measure the bias and mean squared error (MSE) of various estimators of the true mean composition. Additionally, we explore the impact of missing values on estimator performance. Our findings reveal that the optimized binary log-ratio mean, a new estimator, exhibits the lowest MSE and bias. It performs well even with up to 70% missing values, in contrast to other classical estimators such as the arithmetic mean or the geometric mean. Applying our approach to the GeoPT database, which contains replicate analyses of igneous rocks from numerous geochemical laboratories, we introduce an outlier detection technique based on the Mahalanobis distance between a laboratory’s logit coordinates and the optimized mean estimate. This enables a probabilistic ranking of laboratories based on the atypicality of their performance. Finally, we offer an accessible R implementation of our findings through the GitHub repository linked to this paper [subject classification numbers: 10 (compositions) 85 (statistics)].

Abstract

We present a new method for generating pure hexahedral meshes for reservoir simulations. The grid is obtained by extruding a quadrangular mesh, using ideas from the latest advances in computational geometry, specifically the generation of semi-structured quadrangular meshes based on global parameterization. Hexahedral elements are automatically constructed to smoothly honor the geometry of input features (domain boundaries, faults, and horizons), thus making it possible to be used for multiple types of physical simulations on the same mesh. The main contributions are as follows: the introduction of a new semi-structured hexahedral meshing workflow producing high-quality meshes for a wide range of fault systems, and the study and definition of weak verticality on triangulated surface meshes. This allows us to design better and more robust algorithms during the extrusion phase along non-vertical faults. We demonstrate (i) the simplicity of using such hexahedral meshes generated using the proposed method for coupled flow-geomechanics simulations with state-of-the-art simulators for reservoir studies, and (ii) the possibility of using such semi-structured hexahedral meshes in commercial structured flow simulators, offering an alternative gridding approach to handle a wider family of fault networks without recourse to the stair-step fault approximation.

The behavior and evolution trajectory of hydrofracture, which show a close relationship with the hydrothermal mineralization process, is greatly influenced by fluid flow and fluid pressure. However, further investigation is needed to achieve an in-depth understanding of the formation and evolution mechanisms behind the link between the rate of fluid pressure development and the occurrence of induced hydrofracture and mineralization process. We considered different fluid pressure development rates as the initial data for a cellular automaton model. With the increase in the fluid pressure increase rates, the corresponding hydrofracture became more focused, changing in scale from a large number of small-scale hydrofractures to a small number of large-scale hydrofractures. Episodes of fluid pressure fluctuation induced by either low or high fluid pressure increase rates were shown to trigger mineral precipitation and further contribute to the generation of strong spatially structured and enriched geochemical patterns. Moreover, the correlation length at the percolation threshold, which is of great significance to the degree and scale of mineralization, increased with the increasing fluid pressure increase rates. It was concluded that computational grids with high fluid pressure increase rates are much more prone to produce enriched geochemical patterns with strong spatial structures than grids with low fluid pressure increase rates owing to a larger correlation length at the percolation threshold. These results suggest that the way of fluid pressure development is a key factor for quantifying the behavior of hydrofracture and mineralization process.

Abstract

In this paper, an innovative approach for enhancing fluid transport modeling in porous media is presented, which finds application in various fields, including subsurface reservoir modeling. Fluid flow models are typically solved numerically by addressing a system of partial differential equations (PDEs) using methods such as finite difference and finite volume. However, these processes can be computationally demanding, particularly when aiming for high precision on a fine scale. Researchers have increasingly turned to machine learning to explore solutions for PDEs in order to improve simulation efficiency. The proposed method combines an adaptive multi-scale strategy with generative adversarial networks (GAN) to increase simulation efficiency on a fine scale. The devised model, called simulation enhancement GAN (SE-GAN), takes coarse-scale simulation results as input and generates fine-scale results in conjunction with the provided petrophysical properties. With this new approach, a deep learning model is trained to map coarse-scale results to fine-scale outcomes, rather than directly solving the fluid flow model. Case studies reveal that SE-GAN can achieve a significant improvement in accuracy while reducing computational time compared to the original fine-scale simulation solver. A comprehensive evaluation of numerical experiments is conducted to elucidate the benefits and limitations of this method. The potential of SE-GAN in accelerating the numerical solver for reservoir simulations is also demonstrated.

The purpose of mineral prospectivity mapping (MPM) is to discover unknown mineral deposits by means of fusing multisource prospecting information. In recent years, with rapid advancements in artificial intelligence, deep learning algorithms (DLAs) as a groundbreaking technique have exhibited outstanding capabilities in geoscience. However, conventional DLAs for MPM face certain challenges in feature extraction and the fusion of multimodal prospecting data. Moreover, opaque DLAs lead to an insufficient understanding of the predictive results by experts. In this study, a dual-branch convolutional neural network (DBCNN) and its post hoc interpretability were jointly constructed to map gold prospectivity in western Henan Province of China. In particular, channel and spatial attention modules were integrated into two branches to complement the respective advantages of multichannel and high spatial prospecting data for MPM. The Shapley additive explanations (SHAP) framework was then adopted to explain the predictive results by exploring the feature contributions. The comparative experiments illustrated that DBCNN can enhance feature representation and fusion abilities to improve the performance of MPM compared to conventional DLAs. The high-probability areas delineated by the DBCNN model exhibited close spatial relevance with known gold deposits, and the SHAP further confirmed the reliability of the predictive result obtained by the DBCNN model, thereby guiding future gold exploration in this study area.