This paper presents a three-dimensional, geometric-mechanical, hierarchical, stochastic model of natural rock fracture systems. In the model, fracture systems are generated through superposition of hierarchically related sets, created via stochastic methods that reflect inherent relationships between fracture system geometry and underlying geologic mechanisms. The model employs Poisson plane and line processes as well as random spatial rotation and translation to represent fracture orientations, intensity, and relations to major geologic structures. The model is implemented in the computer program GEOFRAC, which incorporates algorithms for representation of fracture systems in different geologic settings, including folds, faults, and central structures. Application of the model to the Permian reservoir in the Yates field in Texas includes geomechanical analysis of fracture evolution, development of case-specific algorithms for fracture intensity modeling based on rock properties, and numerical simulations of fracture sets related to regional depositional trends and reservoir anticlinal structure.


Natural rock fracture systems are three-dimensional (3D) networks of multiple interconnected fractures that evolve under time-and-space-variant geologic stresses. Since field sampling methods of fractures are typically one-dimensional (logs, cores) or two-dimensional (outcrop maps), there is usually great uncertainty about the 3D fracture system geometry.

A 3D model [1], which continues a long tradition in 3D fracture system modeling at the Massachusetts Institute of Technology [2, 3, 4, 5], accounts for that uncertainty through geology-based mathematical and numerical algorithms. The geometric-mechanical model explores the inherent relations between the 3D geometry of fracture systems and the underlying geologic mechanisms. Poisson plane and line processes [6, 7] and random spatial rotation and translation represent orientations and intensity within a fracture set. The UNIX-based C++ code GEOFRAC implements the 3D stochastic model and incorporates routines for generation of fracture systems via superposition of hierarchically related fracture sets in different geologic settings.

This paper presents the fundamentals of the 3D model, and its application to the fracture system in the petroleum reservoir of the Yates field in Texas.


  • A Poisson plane network;

  • Subdivision of planes into fractured and intact areas through Poisson line tessellation and random marking of polygons;

  • Random 3D translation and/or rotation of fractured polygons.

In the 3D model, fractures are convex polygons that are randomly generated as members of fracture sets through three stochastic processes (Figure 1): A fracture set is generated in a modeling volume enclosed by representative surfaces, e.g. bedding planes, structural boundaries, datum planes, and the ground surface. The three stochastic processes reproduce fracture orientations and intensity as they vary within a fracture set, as follows.

2.1. Modeling of stress field orientation: primary stochastic process

The primary stochastic process (Figure 1a): a homogeneous, anisotropic, Poisson plane network [2, 6], represents stress field orientation. The mean orientation of a fracture set is specified in polar coordinates (azimuth, T, and latitude, F) in a global frame of reference (OXYZ), the axes of which coincide with relevant global directions (Figure 2).

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