Elsevier

Computers & Geosciences

Volume 72, November 2014, Pages 1-17
Computers & Geosciences

Review
iBem3D, a three-dimensional iterative boundary element method using angular dislocations for modeling geologic structures

https://doi.org/10.1016/j.cageo.2014.06.007Get rights and content

Highlights

  • Modeling of perturbed stress field around complex 3D frictional discontinuities.

  • Extension to slip inversion using complex fault geometry.

  • Determination of paleostress using any type of data.

  • Application to fracture mechanics, earthquakes, volcanos and risk assessment.

Abstract

Most analytical solutions to engineering or geological problems are limited to simple geometries. For example, analytical solutions have been found to solve for stresses around a circular hole in a plate. To solve more complex problems, mathematicians and engineers have developed powerful computer-aided numerical methods, which can be categorized into two main types: differential methods and integral methods. The finite element method (FEM) is a differential method that was developed in the 1950s and is one of the most commonly used numerical methods today. Since its development, other differential methods, including the boundary element method (BEM), have been developed to solve different types of problems. The purpose of this paper is to describe iBem3D, formally called Poly3D, a C++ and modular 3D boundary element computer program based on the theory of angular dislocations for modeling three-dimensional (3D) discontinuities in an elastic, heterogeneous, isotropic whole- or half-space. After 20 years and more than 150 scientific publications, we present in detail the formulation behind this method, its enhancements over the years as well as some important applications in several domains of the geosciences. The main advantage of using this formulation, for describing geological objects such as faults, resides in the possibility of modeling complex geometries without gaps and overlaps between adjacent triangular dislocation elements, which is a significant shortcoming for models using rectangular dislocation elements. Reliability, speed, simplicity, and accuracy are enhanced in the latest version of the computer code. Industrial applications include subseismic fault modeling, fractured reservoir modeling, interpretation and validation of fault connectivity and reservoir compartmentalization, depleted area and fault reactivation, and pressurized wellbore stability. Academic applications include earthquake and volcano monitoring, hazard mitigation, and slope stability modeling.

Introduction

The rapid increase in the number of geologic, seismologic, and geodetic datasets with abundant and very precise spatial information on fault geometry and slip distributions promotes the development of more complex geometric and kinematic models of modern earthquake ruptures and paleoseismic events. These datasets indicate that faults commonly are composed of multiple discrete segments, each with a curved surface and curved tipline. Construction of model fault segments using multiple rectangular dislocations (Okada, 1985) introduces nonphysical gaps and overlaps with associated stress concentrations and irregularities in slip distributions that may differ significantly from those in nature (Maerten et al., 2005). Discretization of fault segments into a set of triangular dislocations enables one to approximate the curviplanar surfaces and curved tiplines to a precision that is consistent with the data (Jeyakumaran et al., 1992, Thomas, 1993, Maerten et al., 2005, Meade, 2007, Maerten et al., 2010, Maerten, 2010a, Maerten, 2010b).

The C computer code that was originally developed at Stanford University by Andrew Thomas (1993) in 1993 was called Poly3D. The idea of using the angular dislocation formalism to construct complex planar dislocations with constant displacement discontinuity was first used by Jeyakumaran et al. (1992). Since then, because of the rapid evolution of computer power and the constant demand for more complex and larger models, a new code has emerged, following the work of Jeyakumaran et al. (1992) for triangular elements. For the new code, iBem3D, the C++ object-oriented language was chosen, and an iterative solver now replaces the older direct solver (Gauss elimination). C++ allows modularity of the code (Maerten and Maerten, 2008, Maerten et al., 2010, Maerten, 2010a) while the iterative solver permits running larger models in a shorter time (Maerten et al., 2010). The strain field, given by the derivatives of the equations for the displacement field provided by Comninou and Dundurs (1975), was entirely rederived by hand for optimization considerations, whereas these derivatives in Poly3D were symbolically derived using a dedicated software. The call to the core equations now runs four times faster. Comparisons of Poly3D and iBem3D are summarized in Fig. 1, where the technological differences are highlighted.

In this paper, we summarize the theory behind iBem3D, along with verifications (Section 2), and present the latest improvements such as the implementation of material heterogeneity, static friction, optimizations, parallelization, linear-slip inversion, and paleostress recovery (Section 3). Finally, academic, research, and industrial applications are discussed in Section 4.

Section snippets

Theory behind iBem3D

The theory of dislocations in elastic materials has been used widely over the past half century to evaluate the displacement, strain, and stress fields around faults in Earth׳s lithosphere. Steketee, 1958b, Steketee, 1958a has discussed this theory and potential applications to geophysical problems in two papers. He reviewed Volterra׳s formulation for the dislocation problem and presented a method for the construction of Green׳s functions for the semi-infinite space containing a surface of

Enhancements to iBem3D

iBem3D incorporates several new enhancements thanks to the C++ object-oriented design.

Applications

Numerical models of rock deformation based on continuum mechanics can provide significant computational tools for the understanding of geologic structures and phenomena in the context of theoretical research, teaching, hydrocarbon exploration and production, as well as civil engineering.

Conclusions

The C computer program Poly3D has been applied to a wide variety of problems in academic and industrial structural geology since 1993, with over 130 published papers. iBem3D provides a new formulation of the 3D problem of multiple triangular dislocations arranged to model faults and fractures in an elastic whole- or half-space using the boundary element method. It offers significant enhancements over Poly3D, including modularity, an iterative solver, greater model size and complexity,

Acknowledgments

The authors thank M. David Barnett and Huajian Gao of Stanford University, Department of Material Sciences for their help in understanding the dislocation theory. Yann Lagalay is also greatly acknowledged for helping us to identify and correct the “shadow effect”. We also thank all the iBem3D and Poly3D users who have contributed for over two decades to making this code more stable by reporting problems. The two anonymous reviewers are also acknowledged.

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