Mathematical Geosciences最新文献

Borehole thermal energy storage (BTES) represents cutting-edge technology harnessing the Earth’s subsurface to store and extract thermal energy for heating and cooling purposes. Achieving optimal performance in BTES systems relies heavily on selecting the right operational parameters. Among these parameters, charging and discharging flow rates play a significant role in determining the amount of heat that can be effectively recovered from the system. In this study, we introduce a genetic algorithm as an optimization tool aimed at fine-tuning these operational parameters within a baseline BTES model. The BTES model was developed using FEFLOW software and simulated over a 3-year period. After each 3-year simulation, the genetic algorithm iteratively adjusted the operational parameters to attain the optimal configuration for maximizing heat recovery from the BTES system. Additional analysis was conducted to explore the impact of BTES system size and borehole spacing on heat recovery. Results indicate that the genetic algorithm effectively optimized parameters, leading to enhanced heat recovery efficiency. Moreover, the scenario studies highlighted that closer borehole spacing correlates with higher recovery efficiency.

In recent years, numerous countries have initiated geochemical survey projects, highlighting the importance of identifying geochemical anomalies for the discovery of potential mineral deposits. In addition, anthropogenic activity, missing or inaccurate data, and overburden can lead to local enrichment or deficiency of elements, resulting in false or weak geochemical anomalies. Simultaneously considering spatial and spectrum information in the data can eliminate spectrum differences caused by the data inaccuracy and enhance weak mineralization anomalies. Therefore, introducing spatial-spectrum models is beneficial for leveraging the strengths of both approaches. This study proposes a two-branch fusion network for extracting spatial-spectrum features from a geochemical survey data cube. The spectrum branch consists of a one-dimensional convolutional neural network (1DCNN) that can be utilized to extract geochemical spectrum information within a single pixel, covering major and trace geochemical elements and accounting for both positive and negative geochemical anomalies. The spatial branch is a superpixel graph convolutional network (SGCN), which is composed of internal and external graph convolutions. The SGCN not only can extract spatial relationships between neighboring pixels and even pixels at a long distance, but also takes into account the anisotropy of mineralization. Furthermore, spatial information can smooth out false geochemical anomalies caused by inaccurate or missing data. A case study was conducted to identify mineralization-related geochemical anomalies and validate the proposed hybrid deep learning model in northwestern Hubei Province, China. Experiments have shown that (1) superpixel segmentation is an effective tool for geochemical anomalies identification, (2) the incorporation of spectrum- and spatial-based methods contributes to the model’s ability to discern anomalies and backgrounds within the geochemical data cube, improving its accuracy in anomaly detection, and (3) the identified anomalous areas provide clues for future mineralization searches.

Optimization of the expected outcome for subsurface reservoir management when the properties of the subsurface model are uncertain can be costly, especially when the outcomes are predicted using a numerical reservoir flow simulator. The high cost is a consequence of the approximation of the expected outcome by the average of the outcomes from an ensemble of reservoir models, each of which may need to be numerically simulated. Instead of computing the sample average approximation of the objective function, some practitioners have computed the objective function evaluated on the “mean model,” that is, the model whose properties are the means of properties of an ensemble of model realizations. Straightforward use of the mean model without correction for bias is completely justified only when the objective function is a linear function of the uncertain properties. In this paper, we show that by choosing an appropriate transformation of the variables before computing the mean, the mean model can sometimes be used for optimization without bias correction. However, because choosing the appropriate transformation may be difficult, we develop a hierarchical bias correction method that is highly efficient for robust optimization. The bias correction method is coupled with an efficient derivative-free optimization algorithm to reduce the number of function evaluations required for optimization. The new approach is demonstrated on two numerical porous flow optimization problems. In the two-dimensional well location problem with 100 ensemble members, a good approximation of the optimal location is obtained in 10 function evaluations, and a slightly better (nearly optimal) solution using bias correction is obtained using 216 function evaluations.

Fault interpretation in geology inherently involves uncertainty, which has driven the need to develop methods to quantify and analyze this uncertainty. This paper introduces a novel framework for this task by integrating graph theory, entropy, and random walk. The proposed approach employs graph theory to mathematically represent a fault network in both map-view and profile sections. By integrating the theory of two-dimensional random walk, the stochastic nature of the fault growth process can be effectively characterized, enabling the development of tailored probability formulations for the fault network through weighted graph theory. In addition, entropy models tailored to the fault network are formulated, providing a solid foundation for uncertainty quantification and analysis. Furthermore, the proposed method employs the principle of increase of entropy to quantitatively assess the uncertainty involved in comparing different fault networks. A case study is presented to demonstrate the practical application in addressing the challenges associated with quantifying, communicating, and analyzing the uncertainty in fault interpretation. The findings obtained in this study suggest that (1) entropy serves as a reliable metric for measuring and communicating the uncertainty in fault interpretation; (2) entropy can be used to estimate the potential numbers of evolutionary paths available for a fault network; and (3) the growth process of a fault network adheres to the principle of increase of entropy, enabling us to utilize entropy to measure the complexity of the fault network and subsequently compare the differences between various fault networks. The results obtained highlight the potential of this approach not only for understanding the geological meaning of uncertainty in fault interpretation but also for enhancing decision-making in related fields.

Faults are crucial subsurface features that significantly influence the mechanical behavior and hydraulic properties of rock masses. Interpreting them from seismic data may lead to various scenarios due to uncertainties arising from limited seismic bandwidth, possible imaging errors, and human interpretation noise. Although methods addressing fault uncertainty exist, only a few of them can produce curved and sub-seismic faults simultaneously while quantitatively honoring seismic images and avoiding anchoring to a reference interpretation. This work uses a mathematical framework of marked point processes to approximate fault networks in two dimensions with a set of line segments. The proposed stochastic model, namely the Candy model, incorporates simple pairwise and nearby connections to capture the interactions between fault segments. The novelty of this approach lies in conditioning the stochastic model using input images of fault probabilities generated by a convolutional neural network. The Metropolis–Hastings algorithm is used to generate various scenarios of fault network configurations, thereby exploring the model space associated with the Candy model and reflecting the uncertainty. Probability level sets constructed from these fault segment configurations provide insights on the obtained realizations and on the model parameters. The empty space function produces a ranking of the generated fault networks against an existing interpretation by testing and quantifying their spatial variability. The approach is applied to two-dimensional sections of seismic data, in the Central North Sea.

The chances of discovering hidden deposits are higher when exploring deeper into known deposits or historic mines, compared to broad-scale regional exploration. Machine learning algorithms and three-dimensional modeling can effectively identify deep targets and provide quantitative predictions of potential resources. This research paper presents a proposed workflow that utilizes random forest algorithms and a three-dimensional model incorporating geological factors such as strata, lithology, alteration, and primary halo to enhance the accuracy of exploration predictions. The study involved collecting 7949 rock samples from 34 boreholes in eight exploration lines at the Xiongcun No. 2 deposit, and performing geochemical analysis calculations on 18 elements. The methodologies employed can be summarized as follows: (1) establishing and preprocessing the geological dataset of the Xiongcun No. 2 deposit, followed by multivariate statistical analysis, (2) delineating primary halo zoning sequences to identify potential mineralization at greater depths, (3) constructing three-dimensional models incorporating geological and geochemical mineralization information, and (4) utilizing the random forest algorithm to extract exploration criteria and quantitatively predict deep exploration targets. The results indicate a significant mineralization located 300 m to the west–northwest of the No. 2 deposit, within the downward extension of the control depth. The three-dimensional model of the target volume reveals the presence of approximately 0.33 million tons of copper (Cu), 7.6 tons of gold (Au), and 22.8 tons of silver (Ag).

The study and exploration of the subsurface requires the construction of geological models. This task can be difficult, especially in complex geological settings, with various unconformities. These models are constructed from seismic or well data, which can be sparse and noisy. In this paper, we propose a new method to compute a stratigraphic function that represents geological layers in arbitrary settings. This function interpolates the data using piecewise quadratic (C^1) Powell–Sabin splines and is regularized via a self-adaptive diffusion scheme. For the discretization, we use Powell–Sabin splines on triangular meshes. Compared to classical interpolation methods, the use of piecewise quadratic splines has two major advantages. First, they have the ability to produce surfaces of higher smoothness and regularity. Second, it is straightforward to discretize high-order smoothness energies like the squared Hessian energy. The regularization is considered as the most challenging part of any implicit modeling approach. Often, existing regularization methods produce inconsistent geological models, in particular for data with high thickness variations. To handle this kind of data, we propose a new scheme in which a diffusion term is introduced and iteratively adapted to the shapes and variations in the data while minimizing the interpolation error.

The identification of mineral deposit footprints by processing geochemical survey data constitutes a crucial stage in mineral exploration because it provides valuable and substantial information for future prospecting endeavors. However, the selection of appropriate pathfinder elements and the recognition of their anomalous patterns for determining metallogenic favorability based on geochemical survey data remain challenging tasks because of the complex interactions among different geochemical elements and the highly nonlinear and heterogeneous characteristics of their spatial distribution patterns. This study investigated the application of a causal discovery algorithm and deep learning models to identify geochemical anomaly patterns associated with mineralization. Using gold-polymetallic deposits in the Edongnan region of China as a case study, stream sediment samples containing concentrations of 39 elements were collected and preprocessed using a centered log-ratio transformation, addressing the closure effect of compositional data. The combination of the synthetic minority oversampling technique, Tomek link algorithm, and causal discovery algorithm to explore the potential associations and influences among geochemical elements provides new insights into the selection of pathfinder elements. Regarding the problem of identifying anomalous spatial distribution patterns in pathfinder elements and considering that the formation of mineral deposits is the result of various geological processes interacting under specific spatiotemporal conditions, we proposed a hybrid deep learning model called VAE-CAPSNET-GAN, which combines a variational autoencoder (VAE), capsule network (CAPSNET), and generative adversarial network (GAN). The model was designed to capture the spatial distribution characteristics of pathfinder elements and the spatial coupling relationships between mineral deposits and geochemical anomalies, enabling the recognition of geochemical anomaly patterns related to mineralization. The results showed that, compared to the VAE model, which also uses reconstruction error as the anomaly detection principle, VAE-CAPSNET-GAN exhibited superior performance in identifying known mineral deposits and delineating anomalous areas aligned more closely with the established metallogenic model. Furthermore, this weakens the impact of overlapping information. Multiple outcomes indicated that an integrated analytical framework combining a causal discovery algorithm with deep learning models can provide valuable clues for further delineating prospects.

Ensemble-based optimization (EnOpt), commonly used in reservoir management, can be seen as a special case of a natural evolution algorithm. Stein’s lemma gives a new interpretation of EnOpt. This interpretation enables us to study EnOpt in the context of general mutation distributions. In this paper, a non-Gaussian generalization of EnOpt (GenOpt) is proposed, where the control gradient is estimated using Stein’s lemma, and the mutation distribution is updated separately via natural evolution. For the multivariate case, a Gaussian copula is used to represent dependencies between the marginals. The correlation matrix is also iteratively optimized. It is shown that using beta distributions as marginals in the GenOpt algorithm addresses the truncation problem that sometimes arises when applying EnOpt on bounded optimization problems. The performance of the proposed optimization algorithm is evaluated on several test cases. The experiments indicate that GenOpt is less dependent on the chosen hyperparameters, and it is able to converge more quickly than EnOpt on a reservoir management test case.

This paper presents a new method for extracting the image features and lineaments related to local extrema of an image or a digital elevation model (DEM) such as ridges and valleys based on the continuous wavelet transform (CWT) of a set of variously illuminated hillshades. The method originates from the principle that a hillshade can exactly reflect the lineaments nearly perpendicular to the illumination direction of the hillshade, but not other ones. The method consists of four steps: (1) preparation of a set of differently illuminated hillshades of the input data, (2) detection of directional edges nearly perpendicular to the illumination direction from each hillshade based on the CWT, (3) a combination of multidirectional edges into an omnidirectional feature image, and (4) identification of lineaments through linkage and linearization of image feature lines. CWT coefficients of each hillshade are used to calculate the gradient and its direction of the hillshade. For each hillshade, directional edge pixels where the gradient direction is parallel to the illumination direction are selectively detected to form accurate and solitary image feature lines related to local extrema of the input data. Directional edges of each hillshade are easily classified into positive and negative edges using the hillshade gradient. As they have similar directions, they are easily linked to form longer line raster objects, which are converted into vector objects, that is, directional lineaments. The multidirectional edges and lineaments given from all the hillshades are combined to form an omnidirectional feature image and a group of omnidirectional lineaments. Its application to real DEMs shows the demonstrated advantages of the proposed method over other methods and the similarity between detected lineaments and fault lines in the study area.