Jacob Skauvold

There has been much interest recently in implicit artificial intelligence (AI)-based approaches for geostatistical facies modeling. New generative machine learning constructions such as latent diffusion models (LDMs) appear to be competitive with traditional geostatistical approaches for facies characterization. Going beyond visual inspection of predictions, this study examines properties of the statistical distribution of samples generated by an LDM trained to generate facies models. The study uses a traditional truncated Gaussian random field (TGRF) model as a reference data-generating process and as the ground truth for benchmarking the LDM results. The distributions of realizations drawn from the LDM and TGRF models are compared using metrics including bias, variance, higher-order statistics, transiograms and Jensen–Shannon divergence for both marginal and joint (volume) distributions. Comparisons are made with and without conditioning on facies observations in wells for both stationary and nonstationary TGRF models with different covariance functions. The observed distributional differences are modest, and LDMs are regarded as a very promising approach here. Even so, some systematic artifacts are observed, such as underrepresentation of variability by the LDM. Moreover, the performance of the LDM is found to be sensitive to the training data.

Creating accurate and geologically realistic reservoir facies based on limited measurements is crucial for field development and reservoir management, especially in the oil and gas sector. Traditional two-point geostatistics, while foundational, often struggle to capture complex geological patterns. Multi-point statistics offers more flexibility, but comes with its own challenges related to pattern configurations and storage limits. With the rise of Generative Adversarial Networks (GANs) and their success in various fields, there has been a shift towards using them for facies generation. However, recent advances in the computer vision domain have shown the superiority of diffusion models over GANs. Motivated by this, a novel Latent Diffusion Model is proposed, which is specifically designed for conditional generation of reservoir facies. The proposed model produces high-fidelity facies realizations that rigorously preserve conditioning data. It significantly outperforms a GAN-based alternative. Our implementation on GitHub: github.com/ML4ITS/Latent-Diffusion-Model-for-Conditional-Reservoir-Facies-Generation

Stochastic facies models based on truncated Gaussian random fields are known for being flexible and well suited to reproduce patterns and features from analogues or conceptual models. In pluri-Gaussian simulation, the number of random fields is theoretically unlimited, which adds flexibility and makes it possible to model a wider range of geological settings. However, the truncation map traditionally used to set up these models quickly becomes unclear when used for higher dimensions. Hence, in practical pluri-Gaussian applications, the number of fields is typically kept as low as two or three. We present a formulation of pluri-Gaussian simulation in which the truncation rule, the function that maps combinations of Gaussian random field values to facies categories, is represented as a particular binary tree. This is used to decouple the fields in the critical Gibbs sampling step of the conditioning process in such a way that we can use multiple lower-dimensional samples instead of a single higher-dimensional sample. The resulting conditioning algorithm scales excellently with the amount of conditioning data and the number of fields. The algorithm accepts a combination of trends and probabilities in the same model setup, which provides additional flexibility in representing varying depositional geometries. We demonstrate the hierarchical pluri-Gaussian simulation with two practical examples. One is based on real data from the Volve oil field in the North Sea. The other combines a large number of synthetic observations with a truncation tree tailored to a more complex geological concept. The choices made when building the truncation tree affect the features of the realizations, especially when it comes to which facies can be in contact and which can overprint each other. This aspect of tree building is discussed in light of the numerical examples given.

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.

The main aim of the GEOPARD project is to increase the geological realism of the facies models used in reservoir characterization workflow by integrating geological rules at the core of the statistical modelling framework. Through the model we define bedset boundaries, capture realistic interfingering patterns between the sand-rich shoreface and the more mud-rich offshore transition zone, and control facies belt thickness and shoreline trajectories. Due to the close relationship between the model parameterization and geological processes, analogue data are utilized to inform the model.

https://arxiv.org/pdf/2311.01968.pdf

Truncated Gaussian fields are a common way of modelling facies, where the correlation structure in the Gaussian field defines a spatial correlation structure for the facies. Plurigaussian simulation takes it further by using several underlying Gaussian fields. This allows more flexibility and makes it possible to model a wider range of geological settings, but conditioning can be difficult. We present a fast and accurate implementation of conditional plurigaussian simulation. Our approach has two key elements. The first is to combine complex truncation rules with input facies probabilities. The truncation rule, which is a function from the Gaussian fields to a facies value, can be represented neatly as a binary truncation tree. This allows for a general representation that includes all the traditional 2D truncation masks. We show how to combine the use of such trees with facies probabilities, even in complicated cases with more than two Gaussian fields. The second key element is correct conditioning to all facies observations, not just transitions, by treating them as inequality constraints on the Gaussian fields. We perform inequality Kriging by replacing these facies observations by synthetic observations of the underlying Gaussian fields. To generate synthetic observations that agree with the target posterior distribution, we use a Gibbs sampler. Since this is a quite slow algorithm, we take certain measures to make the calculations faster. Synthetic observations are then used in Kriging, improving the conditioning to facies logs from wells. We demonstrate the method with a synthetic case that combines a large number of observations with the use of a truncation tree tailored from a geological concept.

Ensemble-based data assimilation methods like the ensemble Kalman filter must estimate covariances between state variables and observed variables to update ensemble members. In high-dimensional, geostatistical estimation settings where the system state consists of spatial random fields, spurious entries in estimated covariance matrices can degrade the predictive performance of posterior ensembles. We propose to avoid spurious correlations by specifying a parametric form for the state covariance, and fitting this model to the forecast ensemble. The idea is demonstrated on a partially synthetic North Sea test case involving forward stratigraphic modeling.