Finite Difference Implicit Structural Modeling of Geological Structures

Abstract

We introduce a new method for implicit structural modeling. The main developments in this paper are the new regularization operators we propose by extending inherent properties of the classic one-dimensional discrete second derivative operator to higher dimensions. The proposed regularization operators discretize naturally on the Cartesian grid using finite differences, owing to the highly symmetric nature of the Cartesian grid. Furthermore, the proposed regularization operators do not require any special treatment on boundary nodes, and their generalization to higher dimensions is straightforward. As a result, the proposed method has the advantage of being simple to implement. Numerical examples show that the proposed method is robust and numerically efficient.

Finite Element Implicit 3D Subsurface Structural Modeling

“…This problem is similar to implicit surface reconstruction techniques [13], but it very often represents several non-intersecting horizons with one single scalar field, and it additionally needs to consider multiple discontinuities (faults) within the modeling domain. Implicit structural modeling has traditionally been separated into two main classes [7,1]: (1) mesh-free methods [2,3,14,6,15,16,17,18] , and (2) mesh-based methods [4,19,5,7,8,20,21,22]. We also acknowledge an increasing interest in methods based on machine learning [23].…”

“…Minimization of curvature is achieved by imposing Equation 3 where the matrix R is assembled by discretizing one of the regularization operators above. Following [20,22], we consider the function ϕ(x) to have zero curvature at the point x if, by definition, the second directional derivative ∂ 2 k ϕ(x) (and hence the curvature) along all directions d k vanishes at that point. In particular, given the Hessian matrix…”

Deep Relative Geologic Time: A Deep Learning Method for Simultaneously Interpreting 3‐D Seismic Horizons and Faults

“…An accurate RGT volume is thus helpful to well understand geologic structures of interest and qualify reservoir properties of continuity and morphology to interpret all associated geologic structures, and to build a subsurface model of a particular environment. It has shown great promise in many applications to improve horizon extraction (Stark, 2004; Fomel, 2010; X. Wu & Zhong, 2012), sedimentologic interpretation (Hongliu et al., 2012), stratigraphic interpretation (Alliez et al., 2007; Karimi & Fomel, 2015; Lacaze et al., 2017), geologic body detection (Prazuck et al., 2015), and structural implicit modeling (Grose et al., 2017; Irakarama et al., 2020; Laurent et al., 2016; Renaudeau et al., 2019).…”

“…It has shown great promise in many applications to improve horizon extraction (Stark, 2004;Fomel, 2010;X. Wu & Zhong, 2012), sedimentologic interpretation (Hongliu et al, 2012), stratigraphic interpretation (Alliez et al, 2007;Lacaze et al, 2017), geologic body detection (Prazuck et al, 2015), and structural implicit modeling (Grose et al, 2017;Irakarama et al, 2020;Laurent et al, 2016;Renaudeau et al, 2019).…”

Modelling of faults in LoopStructural 1.0

“…This challenge exists because many of the observations and structural geology rules are not directly integrated into the mathematical description of the geological models. Structural geologists typically use their knowledge and expertise to formulate a working hypothesis in a digital format, which often derives from a biased conceptual mental picture (Jessell et al, 2014). The modelled geometries will always converge towards the geologist's bias and conceptual model, meaning that model building cannot be used to test different geological hypotheses.…”