The Direct Sampling method to perform multiple‐point geostatistical simulations

TLDR

Multiple‑point geostatistics models spatial fields with complex structures and enables control of connectivity patterns critical for groundwater flow and transport. The study proposes sampling directly from the training image for a data event, eliminating the need for a database. The method uses data events from a training image, traditionally stored in a database for conditional probabilities, but instead samples directly from the image, and can simulate geological heterogeneity with or without indirect observations. The method is statistically equivalent to prior approaches yet extends multiple‑point geostatistics to continuous and multivariate variables, works with complex features, nonlinear relationships, and nonstationarity, and is fast, parallelizable, memory‑efficient, and easy to implement.

Abstract

Multiple‐point geostatistics is a general statistical framework to model spatial fields displaying a wide range of complex structures. In particular, it allows controlling connectivity patterns that have a critical importance for groundwater flow and transport problems. This approach involves considering data events (spatial arrangements of values) derived from a training image (TI). All data events found in the TI are usually stored in a database, which is used to retrieve conditional probabilities for the simulation. Instead, we propose to sample directly the training image for a given data event, making the database unnecessary. Our method is statistically equivalent to previous implementations, but in addition it allows extending the application of multiple‐point geostatistics to continuous variables and to multivariate problems. The method can be used for the simulation of geological heterogeneity, accounting or not for indirect observations such as geophysics. We show its applicability in the presence of complex features, nonlinear relationships between variables, and with various cases of nonstationarity. Computationally, it is fast, easy to parallelize, parsimonious in memory needs, and straightforward to implement.