International Journal of Rock Mechanics and Mining Sciences
Journal Review ArticleA review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering☆
Introduction
The purpose of this Journal Review Article is to present the techniques, advances, problems and likely future developments in numerical modelling for rock mechanics. In this Section, the review is prefaced by noting the special nature and idiosyncracies of rock masses—and hence some of the difficulties associated with capturing the rock reality in the numerical models. The utility of numerical modelling in providing understanding for rock engineering design and construction is also explained. Finally, we note here the scope and content of the Review with its emphasis on summarizing trends and providing an extensive literature source.
The reason for the general difficulty in modelling rock masses, by whatever numerical method, is that rock is a natural geological material, and so the physical or engineering properties have to be established, rather than defined through a manufacturing process. The rock mass is largely Discontinuous, Anisotropic, Inhomogeneous and Not-Elastic (DIANE), (Harrison and Hudson, 2000) [1]. Rock masses are under stress and continuously loaded by dynamic movements of the upper crust of the Earth, such as tectonic movements, earthquakes, land uplifting/subsidence, glaciation cycles and tides. A rock mass is also a fractured porous medium containing fluids in either liquid or gas phases, e.g. water, oil, natural gases and air, under complex in situ conditions of stresses, temperature and fluid pressures. The complex combination of constituents and its long history of formation make rock masses a difficult material for mathematical representation via numerical modelling.
In relation to the generally discontinuous nature of rock masses, the photograph of a blasted rock surface in Fig. 1 highlights the fact that rock masses contain through-going pre-existing fractures,1 as well as fractures introduced by the excavation process.
Most of the fractures visible in Fig. 1 are pre-existing natural fractures. Although these rock fractures have occurred naturally through geological processes, their formation is governed by mechanical principles, as illustrated by the three main sets of fractures that, in this case, are mutually orthogonal and divide the rock mass into cuboids. The fractures are most often clustered in certain directions resulting from their geological modes and history of formation. One of the main tasks of numerical modelling in rock mechanics is to be able to characterize such mechanical discontinuities in a computer model—either explicitly or implicitly—the so-called ‘material conceptualization’. Additionally, the interaction between the rock mass and the engineering structure has to be incorporated in the modelling procedure for design, so that consequences of the construction process have also to be characterized.
To adequately represent the rock mass in computational models, capturing such fracturing and the complete DIANE nature of the rock mass, plus the consequences of engineering, it is necessary to be able to include the following features during model conceptualization:
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the relevant physical processes and their mathematical representations by partial differential equations (PDEs), especially when coupled thermal, hydraulic and mechanical processes need to be considered simultaneously;
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the relevant mechanisms and constitutive laws with the associated variables and parameters;
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the pre-existing state of rock stress (the rock mass being already under stress);
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the pre-existing state of temperature and water pressure (the rock mass is porous, fractured, and heated by a natural geothermal heat gradient or man-made heat sources)
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the presence of natural fractures (the rock mass is discontinuous);
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variations in properties at different locations (the rock mass is inhomogeneous);
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variations of properties in different directions (the rock mass is anisotropic);
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time/rate-dependent behaviour (the rock mass is not elastic and may undergo creep or plastic deformation);
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variations of properties at different scales (the rock mass is scale-dependent);
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the effects resulting from the engineering perturbations (the geometry is altered).
The extent to which these features can actually be incorporated into a computer model will depend on the physical processes involved and the modelling technique used; hence, both the modelling and any subsequent rock engineering design will contain subjective judgements.
Rock engineering projects are becoming larger and more demanding in terms of the modelling requirements, one of which, for example, may be to include coupled thermo-hydro-mechanical (THM) behaviour into the model. A truly fully coupled model (including extra processes, such as chemistry) requires complete knowledge of the geometrical and physical properties and parameters of the fractured rock masses. Thus, the challenge is to know how to develop an adequate model. The model does not have to be complete and perfect: it only has to be adequate for the purpose.
For these reasons, rock mechanics modelling and rock engineering design are both a science and an art. They rest on a scientific foundation but require empirical judgements supported by accumulated experiences through long-term practices. This is the case because the quantity and quality of the supporting data for rock engineering design and analysis can never be complete, even though they can be perfectly defined in models.
Some form of predictive capability is necessary in order to coherently design an engineered structure, whether it be on the rock mass surface or within the underground rock mass, and whether it be for civil engineering addressed in this CivilZone review or for mining, petroleum or environmental engineering. The predictive capability is achieved through a variety of modelling methods. Even if one simply adopts the same design as a previously constructed structure, the rock mass condition is generally site-specific and one should use a computer model adopted for the specific site conditions to ensure that the rock mass is likely to behave in similar fashion.
As rock mechanics modelling has developed for the design of rock engineering structures with widely different purposes, and because different modelling methods have been developed, we now have a wide spectrum of modelling approaches. These can be presented in different ways: the categorization into eight approaches based on four methods and two levels, as illustrated in Fig. 2, is from (Hudson, 2001) [2].
The modelling and design work starts with the objective, the top box in Fig. 2. Then there are the eight modelling and design methods in the main central box. The four columns represent the four main modelling methods:
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Method A: Design based on previous design experiences,
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Method B: Design based on simplified models,
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Method C: Design based on modelling which attempts to capture most relevant mechanisms, and
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Method D: Design based on ‘all-encompassing’ modelling.
There are two rows in the large central box in Fig. 2. The top row, Level 1, includes methods in which there is an attempt to achieve one-to-one mechanism mapping in the model. In other words, a mechanism which is thought to be occurring in the rock reality and which is to be included in the model is modelled directly, such as explicit stress–strain relations. Conversely, the lower row, Level 2, includes methods in which such mechanism mapping is not direct. The consequences of, for example, the constitutive models and associated parameters may well be contained within the four modelling and design methods in Level 2, but one cannot explicitly identify the relation within the methodologies, e.g. in the rock mass classification techniques.
Some supporting rock mass characterization parameters will be obtained from site investigation, the left-hand box. Then the rock engineering design and construction proceeds, with a feedback loop to the modelling from construction.
An important point is that in rock mechanics and engineering design, having insufficient data is a way of life, rather than a simple local difficulty, and that is why the empirical approaches (i.e. classification systems) have been developed and are still required. Therefore, we will also be discussing the subject of parameter representability associated with sample size, representative elemental volume (REV), homogenization/upscaling, because these are fundamental problems associated with modelling, and are relevant to the ABCD method categories in Fig. 2.
The use of computers makes significant contributions to all the eight modelling and design methods in Fig. 2; however, the specific numerical methods and approaches that are being reviewed here are used directly in Methods 1C and 1D. Also, there is concentration on the actual numerical methods (rather than computing per se or design per se) and discussion on the rock mass characterization issues related to the numerical methods. Highlighted are the techniques, advances, coupled mechanisms, technical auditing and the ability to present the content of the modelling, the outstanding issues, and the future of this type of modelling. In short, highlighted is the special contribution that numerical models are currently making to rock mechanics.
Because the focus of this Review is on the modelling concepts, the associated special features of modelling rock fractures, the main development milestones, typical application requirements, development trends, and outstanding issues of importance and difficulty, special attention is paid to Section 3 for alternative formulations in each of the modelling methods, noting the potentials for rock mechanics problems. It is hoped that this treatment will provide readers with a comprehensive presentation of the state-of-the-art of numerical analysis in rock mechanics in general, and civil engineering applications in particular—in terms of historical background, presents status and likely future trends.
Section snippets
Numerical methods in rock mechanics
Before considering the details and advances in the specific numerical modelling methods (presented in Section 3), an introduction is provided here to the methods and there is discussion on the continuum vs. discrete approaches. Also considered is the characterization of rock masses which is necessary to provide input to the numerical models, and there is illustration of how enhanced understanding is obtained through the use of such models.
Basic concepts
The FDM is the oldest numerical method to obtain approximate solutions to PDEs in engineering, especially in fluid dynamics, heat transfer and solid mechanics. The basic concept of FDM is to replace the partial derivatives of the objective function (e.g. displacement) by differences defined over certain spatial intervals in the coordinate directions, Δx, Δy, Δz, which yields a system of algebraic simultaneous equations of the objective functions at a grid (mesh) of nodes over the domain of
Coupled thermo-hydro-mechanical models
The couplings between the processes of heat transfer, fluid flow and stress/deformation in fractured rocks has become an increasingly important subject in rock mechanics and engineering design since the early 1980s (Tsang, 1987, 1991) [678], [679], mainly due to the modelling requirements for the design and performance assessment of underground radioactive waste repositories, and other engineering fields in which heat and fluids play important roles, such as gas/oil recovery, hot-dry-rock
Inverse solution methods and applications
A large and very important class of numerical methods in rock mechanics and civil engineering practice is the inverse solution techniques. The essence of the inverse solution approach is to identify unknown system properties or perturbation parameters, through direct application of numerical methods or closed-form solutions to derive unknown material properties, system geometry, and boundary or initial conditions, based on a limited number of measured values of some key variables, using either
Advances and outstanding issues
This Review began by describing the special features of rock masses, the difficulties of characterizing their DIANE nature, and presenting an overview diagram of rock mechanics modelling approaches. The various numerical modelling techniques were then described in some detail with comments on their applicability, followed by a Section on the way in which coupled models are being developed. It has also been intimated that, after the past 50 years’ development, we still mainly use empirical
Conclusions
The conclusions are presented in two parts: firstly, specific conclusions relating to the main numerical modelling methods; and then overview comments relating to the general subject of numerical modelling in rock mechanics.
Acknowledgements
I would like to express my sincere appreciation and gratitude to Professor B.H.G. Brady, Professor Y. Ohnishi, Professor W.G. Pariseau and Dr. R.W. Zimmerman for their comments, suggestions, corrections and especially encouragement in their reviews of this paper. Special thanks to Professor J.A. Hudson who contributed substantially to this review, especially the first two sections and the section about the neural networks, and insisted on removing his name as the co-author.
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This is the second of a series of Journal Review Articles commissioned by the Editor. The series consists of articles reviewing significant or topical subjects, or subjects requiring expert explanation. This Review is a significantly expanded version of the “Numerical Methods in Rock Mechanics” CivilZone Review paper which was published in Vol. 39, No. 4, June 2002, pp. 409–427, and is longer than usual papers in order to do justice to the subject. Also, an enhanced referencing system has been used here with the references being provided in two ways: firstly, by author and date, so that this information is contained directly in the text; and, secondly, by bracketted numbers, following the standard Journal format.