Implicit Modelling of Geological Structures: A Cartesian Grid Method Handling Discontinuities With Ghost Points

Abstract

In geology, implicit structural modelling constructs the geometry of geological structures (e.g. layers) by interpolating between sparse field data. A model is represented by a volumetric scalar field which is discontinuous on structural discontinuities such as faults or stratigraphic unconformities. The management of such discontinuities may involve boolean operations on several scalar fields or the creation of conformal meshes. Instead, we propose a ghost cell technique for the cartesian grid together with a set of relations between the ghost points, the regular nodes, and the discontinuities. Consequently, poor quality meshes are avoided and only the resolution of the grid has an impact on the solution. The modelling problem is posed as a least square's minimization of a bending energy penalization on data mismatch functions and approximated by finite difference. As all relationships in the grid are implicitly defined, except close to the discontinuities, this algorithm is computationally efficient. We provide some benchmarks of the method on two-dimensional examples with folds, faults, and erosions.

Finite Difference Implicit Structural Modeling of Geological Structures

“…This can dramatically increase both the computational time and the computer memory required to solve the system depending on how close the faults are. One strategy to address this resolution limitation in finite difference implicit modeling would be to use additional degrees of freedoms on nodes adjacent to faults as done for example in the extended finite element method (Moes and Belytscheko, 2002), and also recently used in structural modeling (Renaudeau et al, 2018).…”

“…Computation times that used to be acceptable for building a single "best model" are becoming far too long when considering stochastic approaches for sampling uncertainty. Other recent research and advances in implicit modeling include better modeling of folds Grose et al, 2017), automated building of models that conform to seismic data (Wu, 2017), more numerically efficient discretization schemes (Renaudeau et al, 2018), and many more (Mallet, 2014;Gonçalves et al, 2017;Martin and Boisvert, 2017;Renaudeau et al, 2019). As we move towards an era of multi-realization structural modeling (Caumon, 2010), new challenges continuously emerge and motivate the quest for more robust and more efficient structural implicit modeling schemes.…”

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].…”

LoopStructural 1.0: time-aware geological modelling

“…Different interpolation algorithms can be used for interpolating these surfaces with the ability to mix and match interpolation algorithms depending on the surface type being modelled. LoopStructural has native implementation of discrete implicit modelling using a piecewise linear interpolation on a tetrahedral mesh (Caumon et al, 2013;Frank et al, 2007;Mallet, 2014Mallet, , 2002, finite-difference interpolation on a Cartesian grid (Irakarama et al, 2018;Renaudeau et al, 2018), fold interpolation using tetrahedral meshes (Laurent et al, 2016) and an interface to a generalised radial basis interpolation (Hillier et al, 2014).…”