Stochastic Pix2pix: A New Machine Learning Method for Geophysical and Well Conditioning of Rule-Based Channel Reservoir Models

Constructing subsurface models that accurately reproduce geological heterogeneity and their associated uncertainty is critical to many geoscience and engineering applications. For hydrocarbon reservoir modeling and forecasting, for example, spatial variability must be consistent with geological processes, geophysical measurements, and time records of fluid production measurements. Generating such subsurface models can be time-consuming; conditioning them to different types of measurements is even more computationally intensive and technically challenging. Using too many free variables can cause overfitting of the data, thereby decreasing the predictive ability of the model; high dimensionality also slows convergence during history matching of fluid production measurements. To contend with these problems, we introduce here a new machine learning approach, referred to as the stochastic pix2pix method, which parameterizes high-dimensional, stochastic reservoir models into low-dimensional Gaussian random variables in latent space. Many of the world’s significant hydrocarbon fields originate from fluvial or turbidite deposits, with sedimentary processes and spatial distribution of rock facies significantly influencing their flow behavior. Rule- or object-based methods are commonly used to model geostatistically these types of reservoirs. Here, we introduce a new and efficient machine learning-based reservoir modeling workflow capable of generating 2D fluvial reservoir models that account for the available field data and the geometries of different facies. By constraining subsurface model realizations to available geophysical and petrophysical interpretations, multiphysics inversion can be greatly accelerated, thereby requiring only production history matching. Although our models are 2D, an extension to 3D can be readily implemented through zone-by-zone (i.e., reservoir unit) modeling and conditioning. The proposed method and workflow also partially solve the common problem of machine learning methods, wherein mapping low- to high-resolution images often yield reduced spatial variability. The proposed method is an extension of conditional generative adversarial networks, in which we incorporate a novel penalty term into the loss function in order to generate various realizations honoring the same conditional data, such as structural interpretation from seismic data, and borehole measurements at key well locations. We “train” the generative model on geologically realistic, multivariate spatial models generated with a rule-based fluvial reservoir simulator. Each high-resolution training image includes five lithofacies with distinct petrophysical property trends. To evaluate the performance of the proposed method, visual inspection, indicator variograms and multiple point density function are applied to gauge how well the realizations reproduce existing patterns in the true model. A new metric, the mean categorical error, is proposed to quantify how well the realizations match the conditioning data. The proposed method correctly reproduces patterns even when the conditioning data are significantly different from those in the training set. Likewise, the method can perform continuous model modifications, meaning that the machine learning procedure effectively reproduces the migration rule of a meandering (fluvial) system.