Combining topology and fractal dimension of fracture networks to characterise structural domains in thrusted limestones

Highlights

• A new method for characterising fractures in fold-and-thrust belts with complex fracture networks.

• Reduced time for data collection compared to traditional techniques.

• Fractal dimensions and topology are combined to characterise fractures in structural domains.

• Fore-thrusts and back-thrusts have higher fractal dimensions than pop-up structures.

• Fore-thrusts have fewer longer fractures, back-thrusts have higher densities of connected fractures.

Abstract

Fractures in limestones of the Palaeocene Lockhart Formation in the hanging wall of the Himalayan Main Boundary Thrust north of Islamabad are examined, and the data analysed using a combination of topology and fractal dimension to characterise fracture patterns and relate them to structural domains. Neither technique alone allows the recognition of the structural domains. However, when considered together for all the fractures within an area, fore-thrusts, pop-ups and back-thrusts can be distinguished. The fractures are considered together, as the characteristics of the individual structural domains are characterised by the cumulative effect of all the different fractures, and in these complexly fractured rocks, the concept of fracture sets is problematic. Fore- and back-thrusts have higher fractal dimensions than pop-up structures. The highest fractal dimensions of both types of thrusts occur immediately adjacent to and decrease away from the central pop-up structure. Topologically, fore-thrust domains have fewer fractures and fracture intersections (nodes), with a longer mean fracture trace length; back-thrust domains contain more nodes (hence also more tips, lines, and branches) resulting in higher fracture densities. Pop-up structure domains are characterised by a low fracture intensity. Using the combined analysis of both the topology and fractal dimension, we show that the fracture pattern characteristics are predictable when related to the different structural settings identified within fold and thrust of the Lockhart Formation.

Introduction

Fracturing of a rock mass is a mechanical response to an applied stress (e.g., Ramsay, 1967; Long et al., 1996), with the extent and characteristics of the resultant fracture network controlled by the mechanical properties of the rock mass, fluid characteristics, and variations in the stress field (e.g., Laubach et al., 2019). Understanding the properties and characteristics of the resultant fracture network is essential in many aspects of applied geoscience, from determining the stability of an excavation (Hoek and Brown, 1980) to identifying fluid pathways and storage volumes for minerals (Cox, 2005) or hydrocarbons (Aydin, 2000).

Fracture systems are defined as geometrical arrays of linked and often interacting fractures within a rock mass (Rouleau and Gale, 1985; Odling et al., 1999). Fracture systems have attracted much scientific attention and numerous methods have been proposed to characterise them, ranging from analysis of their kinematic behaviour, through shared and/or discrete geometry, to tectonic setting, as concisely and instructively summarised by Peacock and Sanderson (2018).

The geometric arrangements of fractures in a rock volume are typically viewed as either discrete objects in space (Barros-Galvis et al., 2015; Welch et al., 2015), or topologically, that is to say, 'in relation to one another' (Long and Witherspoon, 1985; Laubach et al., 2018), and/or in direct relation to causative mechanisms.

Studies that consider the spatial distribution of fractures as discrete objects provide valuable insights into the relationships between fractures and lithological characteristics of the fractured rock mass. For example, the spacing of fractures commonly varies with lithology or, more correctly, with differences in the mechanical properties of the lithology, such that competent lithologies display more widely-spaced fractures, for a given stress, compared to their less competent counterparts (Pollard and Fletcher, 2005; Ortega et al., 2010; Hooker et al., 2013). Fracture spacing also varies with bed thickness (Ladeira and Price, 1981) with thicker beds containing more widely-spaced fractures than their thinner equivalents, for a given stress. In folded strata, differences in the geometry of fracture patterns are related to variations in competence and bed thickness and a response to the complex strain distribution in fold systems. This results in a broad array of geometrical fracture characteristics associated with ductile/brittle-ductile fold deformation features (Cosgrove, 2015; Ferrill et al., 2016).

By contrast, topological analysis of a fracture network characterises the connectivity of the constitutive fractures in that network, rather than the inherent properties of the individual fractures (Sanderson et al., 2019). This approach has provided an improved understanding of the overall behaviour of the physical properties of the rock mass under consideration, particularly in terms of its strength, porosity, and permeability (Sanderson and Nixon, 2015).

Approaches to fracture characterisation that establish a causative relationship between a particular fracture system and the mechanism responsible for its formation require observations that can indicate a temporal link between a fracture network and the proposed process (Long et al., 1996). Examples include studies of how fracture systems of different ages (established by geochemistry) link together to control mineralisation within Archean orogenic gold (Dziggel et al., 2007) or recognition of mining-induced fractures and pre-existing geological discontinuities and how they interact to produce the rock mass around a mining stope (Grodner, 1999).

The task of relating a fracture system to a specific process is particularly challenging for rocks that have been subjected to multiple deformational events. For example, in fold-and-thrust belts deformation results from a combination of burial, changes in fluid pressure and composition, folding, thrusting, uplift and exhumation (Engelder, 1985; English and Laubach, 2017). The distribution of fractures variously reflects the different failure responses to stresses of these events due to variations in mechanical properties of the rock mass (Wennberg et al., 2006), that themselves evolve through time (Laubach et al., 2009). Progressive folding can also result in multiple generations of opening-mode fractures (Cosgrove, 2015). Consequently, polyphase deformation in fold-and-thrust belts typically results in complex, sequential overlays of fracture networks with such high abundances and intricate patterns that they are not readily described by simple fold-fault-fracture geometries (Cosgrove, 2015), or by one-dimensional descriptors (Watkins et al., 2015; Laubach et al., 2018). Fractures formed at the same time can have different orientations and mineral compositions and conversely fractures formed at different times can have the same orientations or mineralisation (Laubach et al., 2019). To properly quantify the effects of the fracture networks on the rock mass, the whole fracture system must be considered rather than apparently discrete fracture sets in a fracture network (Peacock et al., 2018).

Here we present a novel approach to the challenges involved in developing an informative, and potentially predictive, characterisation of highly fractured rock. The individual constituent fracture types within the fracture system are not separated for analysis, but rather we consider how the cumulative effects can be used to discriminate different structural domains. This approach integrates discrete topological and spatial methods for characterising fractures and fracture networks by employing fractal dimension to provide a spatial context of the distribution of the constituent fractures, and then combining those data with analyses of the observed topological relationships and interconnectivity of the fracture networks. The approach provides a more robust assessment and analysis of the fractures observed within the rock mass and their characteristics than can be achieved from application of either method in isolation. As we consider all the topological and fractal data together, all the interactions between fractures, and their effects upon the characteristics of the rock mass are defined. Moreover, this approach dramatically reduces the time taken for data collection compared to traditional fracture sampling techniques and provides large amounts of unbiased data representative of fracture network characteristics over a wide range of fracture structural domains.

We apply this technique to examine the occurrence and distribution of fracturing in well-exposed in Palaeocene limestones within the frontal thrust sheets associated with the Main Boundary Thrust (MBT) of the Himalayan fold and thrust belt (Tariq et al., 2017; Dasti et al., 2018), in a region approximately 10 km north of Islamabad, NW Pakistan (Fig. 1 and Fig. 2). Here, in a single stratigraphic unit (the Lockhart Limestone) a complex sequence of fractures can be studied across fore-thrusts, back-thrusts, and pop-up structures that all occur above, and immediately to the north of the MBT. We recognise that there are multiple generations of fractures in the study area, but as the geomechanical properties of the rock mass must be the result of all fractures combined, we contend that it is important to consider all fractures collectively to understand differences in the cumulative distribution of fracture sets related to specific structures. Restricting the structural analysis to a single stratigraphic unit removes variation in fracture characteristics related to lithology.