In many earth sciences applications, the geological objects or structures to be reproduced are curvilinear, e.g., sand channels in a clastic reservoir. Their modeling requires multiple-point statistics involving jointly three or more points at a time, much beyond the traditional two-point variogram statistics. Actual data from the field being modeled, particularly if it is subsurface, are rarely enough to allow inference of such multiple-point statistics. The approach proposed in this paper consists of borrowing the required multiple-point statistics from training images depicting the expected patterns of geological heterogeneities. Several training images can be used, reflecting different scales of variability and styles of heterogeneities. The multiple-point statistics inferred from these training image(s) are exported to the geostatistical numerical model where they are anchored to the actual data, both hard and soft, in a sequential simulation mode. The algorithm and code developed are tested for the simulation of a fluvial hydrocarbon reservoir with meandering channels. The methodology proposed appears to be simple (multiple-point statistics are scanned directly from training images), general (any type of random geometry can be considered), and fast enough to handle large 3D simulation grids.

Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics

Monte Carlo methods have become important in analysis of nonlinear inverse problems where no analytical expression for the forward relation between data and model parameters is available, and where linearization is unsuccessful. In such cases a direct mathematical treatment is impossible, but the forward relation materializes itself as an algorithm allowing data to be calculated for any given model. Monte Carlo methods can be divided into two categories: the sampling methods and the optimization methods. Monte Carlo sampling is useful when the space of feasible solutions is to be explored, and measures of resolution and uncertainty of solution are needed. The Metropolis algorithm and the Gibbs sampler are the most widely used Monte Carlo samplers for this purpose, but these methods can be refined and supplemented in various ways of which the neighbourhood algorithm is a notable example. Monte Carlo optimization methods are powerful tools when searching for globally optimal solutions amongst numerous local optima. Simulated annealing and genetic algorithms have shown their strength in this respect, but they suffer from the same fundamental problem as the Monte Carlo sampling methods: no provably optimal strategy for tuning these methods to a given problem has been found, only a number of approximate methods.

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Monte Carlo analysis of inverse problems

The advent of multiple-point geostatistics (MPS) gave rise to the integration of complex subsurface geological structures and features into the model by the concept of training images Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction In these approaches, the training image is used to construct a pattern database Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image In this paper, an entirely different approach will be taken toward geostatistical modeling A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented In the developed methodology, patterns scanned from the training image are represented as points in a Cartesian space using multidimensional scaling The idea behind this mapping is to use distance functions as a tool for analyzing variability between all the patterns in a training image These distance functions can be tailored to the application at hand Next, by significantly reducing the dimensionality of the problem and using kernel space mapping, an improved pattern classification algorithm is obtained This paper discusses the various implementation details to accomplish these ideas Several examples are presented and a qualitative comparison is made with previous methods An improved pattern continuity and data-conditioning capability is observed in the generated realizations for both continuous and categorical variables We show how the proposed methodology is much less sensitive to the user-provided parameters, and at the same time has the potential to reduce computational time significantly

Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

Whereas outcrop models can provide important information on reservoir architecture and heterogeneity, it is not entirely clear how such information can be used exhaustively in geostatistical reservoir modeling. Traditional, variogram-based geostatistics is inadequate in that regard because the variogram is too limiting in capturing geologic heterogeneity from outcrops. A new field, termed multiple-point geostatistics, does not rely on variogram models. Instead, it allows capturing structure from so-called training images. Multiple-point geostatistics borrows multiple-point patterns from the training image, then anchors them to subsurface well-log, seismic, and production data. However, multiple-point geostatistics does not escape from the same principles as traditional variogram-based geostatistics: it is still a stochastic method and, hence, relies on the commonly forgotten principles of stationarity and ergodicity. These principles dictate that the training image used in multiple-point geostatistics cannot be chosen arbitrarily and that not all outcrops might be suitable training image models. In this paper, we outline the guiding principles of using analog models in multiple-point geostatistics and show that simple, so-called modular training images can be used to build complex reservoir models using geostatistics algorithms.

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Multiple-point Geostatistics: A Quantitative Vehicle for Integrating Geologic Analogs into Multiple Reservoir Models

FLUVSIM: a program for object-based stochastic modeling of fluvial depositional systems

This paper describes a novel approach to modeling braided stream fluvial reservoirs. The approach is based on a hierarchical set of coordinate transformations involving relative straingraphic coordinates, translations, rotations, and straightening functions. The emphasis is placed on geologically sound geometric concepts and realistically-attainable conditioning statistics including areal and vertical facies proportions. Modeling proceeds in a hierarchical fashion, that is (1) a stratigraphic coordinate system is established for each reservoir layer, (2) a number of channel complexes are positioned within each layer, and then (3) channels are positioned within each channel complex. The geometric specification of each sand-filled channel within the background of floodplain shales is a marked point process. Each channel is marked with a starting location, size parameters, and sinuosity parameters. We present the hierarchy of eight coordinate transformations, introduce an analytical expression for the channel cross-section shape, describe the simulation algorithm, and demonstrate how the realizations are made to honor local conditioning data from wells and global conditioning data such as areal and vertical proportions.

Hierarchical object-based stochastic modeling of fluvial reservoirs

Large-scale reservoir architecture constitutes first-order reservoir heterogeneity and dietates to a large extent reservoir flow behavior. It also manifests geometric characteristics beyond the capability of traditional geostatistical models conditioned only on single-point and two-point statistics. Multiple-point statistics, as obtained by scanning a training image deemed representative of the actual reservoir, if reproduced properly provides stochastic models that better capture the essence of the heterogeneity. A “growth” algorithm, coupled with an optimization procedure, is proposed to reproduce target multiple-point histograms. The growth algorithm makes an analogy between geological accretion process and stochastic process and amounts to restricting the random path of sequential simulation at any given stage to a set of eligible nodes (immediately adjacent to a previously simulated node or sand grain). The proposed algorithm, combined with a multiple-grid approach, is shown to reproduce effectively the geometric essence of complex training images exhibiting long-range and curvilinear structures. Also, by avoiding a rigorous search for global minimum and accepting local minima, the proposed algorithm improves CPU time over traditional optimization procedures by several orders of magnitude. Average flow responses run on simulated realizations are shown to bracket correctly the reference responses of the training image.

Modeling complex reservoir geometries with multiple-point statistics

Chlorophyll-a concentrations in water bodies are one of the most important environmental evaluation indicators in monitoring the water environment. Small water bodies include headwater streams, springs, ditches, flushes, small lakes, and ponds, which represent important freshwater resources. However, the relatively narrow and fragmented nature of small water bodies makes it difficult to monitor chlorophyll-a via medium-resolution remote sensing. In the present study, we first fused Gaofen-6 (a new Chinese satellite) images to obtain 2 m resolution images with 8 bands, which was approved as a good data source for Chlorophyll-a monitoring in small water bodies as Sentinel-2. Further, we compared five semi-empirical and four machine learning models to estimate chlorophyll-a concentrations via simulated reflectance using fused Gaofen-6 and Sentinel-2 spectral response function. The results showed that the extreme gradient boosting tree model (one of the machine learning models) is the most accurate. The mean relative error (MRE) was 9.03%, and the root-mean-square error (RMSE) was 4.5 mg/m3 for the Sentinel-2 sensor, while for the fused Gaofen-6 image, MRE was 6.73%, and RMSE was 3.26 mg/m3. Thus, both fused Gaofen-6 and Sentinel-2 could estimate the chlorophyll-a concentrations in small water bodies. Since the fused Gaofen-6 exhibited a higher spatial resolution and Sentinel-2 exhibited a higher temporal resolution.

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Estimation of Chlorophyll-a Concentrations in Small Water Bodies: Comparison of Fused Gaofen-6 and Sentinel-2 Sensors

Dynamic beam hopping (DBH) is a crucial technology for adapting to the flexibility of service configurations in multi-beam satellite (MBS) communications. To solve the problem that the traditional beam-hopping method is difficult to adapt to the dynamic satellite environment, Deep Reinforcement Learning (DRL) was proposed. DRL can use the previous information to make decisions at the current moment and thus has low time complexity. However, because the generalization ability of the existing DRL methods cannot fully meet dynamic changes in multi-beam satellite communications scenarios, it is difficult to obtain the optimal decision. To address this issue, this letter explores a new framework in which DRL-Powered genetic algorithm (GA) approach. The proposed method uses DRL to assist the decision of DBH. Comparing the proposed method with DRL and traditional GA, simulation results show that the proposed method has a better performance in throughput and fairness when adapting to changes in the satellite environment.

Dynamic Beam Hopping for DVB-S2X GEO Satellite: A DRL-Powered GA Approach

The thermal conductivity ( κ $\kappa $ ) and thermal diffusivity ( D $D$ ) of talc have been measured over a range of temperature (298–1,373 K) and pressure (0.5–3.0 GPa) conditions using the transient plane‐source method. The results show that both the thermal conductivity and thermal diffusivity are dependent upon the prevailing temperature and pressure conditions to a certain extent. As the temperature and pressure increase, the thermal diffusivity monotonically decreases, while the thermal conductivity initially decreases between 298 and 973 K and then increases from 973 to 1,173 K. At low temperatures, phonon scattering is the dominant mechanism for heat transfer; at higher temperatures, photon radiation and dehydration become more prevalent. At temperatures greater than 1,173 K, the thermal conductivity decreases significantly due to aqueous liquid accumulation. Talc may be the cause of the high geothermal gradient in the hot subduction zone.

Libing Wang

Papers

Hierarchical object-based stochastic modeling of fluvial reservoirs

Modeling complex reservoir geometries with multiple-point statistics

Estimation of Chlorophyll-a Concentrations in Small Water Bodies: Comparison of Fused Gaofen-6 and Sentinel-2 Sensors

Dynamic Beam Hopping for DVB-S2X GEO Satellite: A DRL-Powered GA Approach

Thermal Conductivity and Thermal Diffusivity of Talc at High Temperature and Pressure With Implications for the Thermal Structure of Subduction Zones