The Young’s Moduli, Poisson’s Ratios and Poroelastic Coefficients of Coals

This report describes the measurement of coal (and rock) properties using core in laboratory tests where the axial and radial (confining) stresses and the fluid pressures are varied. The results derived from this work show quite different material behaviour to those generally considered to apply. In total this work has been based on tests of 13 coal samples involving the collection of information from some 1200 loading states. The report also makes reference to tests conducted on sedimentary rocks.

This work has been conducted because the key to understanding the mechanical behaviour of any structure (including those in the ground) is a proper knowledge of the relationship between stress and strain in three dimensions of the materials involved. In the case of fractured or porous material it also needs to take account of the effects of fluid pressure within it. These relationships may be described as the constitutive equations and should extend to failure and beyond. The material behaviour may include linearly elastic, nonlinearly elastic, viscous or plastic behaviour and may be described by multiple equations. Anisotropy should be considered where it exists but, owing to the complexity of quantifying it, is frequently unduly simplified or even ignored.  

The model that has dominated most analytical work has been one of isotropic, linearly elastic behaviour. This simplifies the equations required for analysis but has meant that every effort has been made to describe the material behaviour in linearly elastic terms. While this is appropriate for metals and for some hard rock, it has been shown in this work, to be materially inadequate for coal and indeed for most sedimentary rock. Not considering the differences between the simplified models and the real behaviour the materials involved is of importance to every aspect of rock mechanics in a coal mine, including pillar design, roof support, outburst behaviour and gas drainage.  

The study of rock mechanics in the civil and mining engineering has traditionally ignored the effects of fluid pressure unless it involves the effect of a fluid in some known, open fracture, as in the limit equilibrium analysis of jointed blocks.  

Unlike rock mechanics, soil mechanics has considered the effects of fluid pressure in effective stress since the work of Terzaghi and Peck (1947). However the analysis of soil behaviour has traditionally been based upon plastic failure, ignoring its stress-strain behaviour. With the advent of numerical models this has started to change, but slowly, principally because of the difficulty in measuring real material properties. Biot and Wills        (1957) did recognise the effects of poroelastic behaviour in their work on soil consolidation but this knowledge was little used, once again because of the difficulties in measuring material properties. .Here the term poroelasticity refers to the way in which changes in fluid pressure change the dimensions of the material within the confines of elastic behaviour. Analysis of settlement has tended to be based on odometer test results which do not separate fluid pressure and material behaviour, elastic or plastic.  

The advent of numerical models, such as finite element analysis, required the use of some constitutive equations and the first choice has been the isotropic, linearly elastic model. This was later modified to model failure, usually by defining failure occurring at some specific state, and then redistributing the load that was shed by failure within the elastic model. Numerical models have advanced and may include anisotropy, though this is generally limited to orthotropic materials. Nonlinearity remains complex and requires very large amounts of computational power and hence is generally not included, and nor are the effects of fluid pressure.  

This report describes the measurement of coal and rock properties in the laboratory. The results show that the behaviour of these materials is generally quite nonlinear, sometimes anisotropic. It has been shown that coal's stiffness typically changes four-fold and sometimes ten-fold with stress state. Some coals exhibit poroelastic behaviour, changing dimension with internal fluid pressure, while others do not. Dimensional change with fluid pressure appears to be much higher at low stresses than at higher stress states.

The nonlinear behaviour of sedimentary rocks is complex to model because it varies widely. While it can be said that the stiffness of the coal or sedimentary rock is dependent on the stress state and generally increases with stress, the way it does so is quite variable. Sometimes the Young's modulus is dependent on the state of stress in the same direction as the stress tensor, while in other cases it is also dependent on the orthogonal stresses. Models of rock behaviour have therefore to be developed from laboratory tests for use on a case by case basis. Current numerical models do not readily enable the use of such models and this must be the subject of future development.  

The key points that have arisen from this study are that:

• Most sedimentary rocks including coals are highly nonlinear in their stiffnesses within the elastic range;
• Many coals are far softer at low stresses and far stiffer at high stresses than is currently accepted;
• Most coals are relatively isotropic in their behaviour on a ply by ply basis; notable variation may be expected between plies;
• The values of Poisson's ratio of the coals is also highly variable. In the analysis of the tests used the values of individual Poisson's ratios may vary from zero to 0.4;
• Some coals change dimension with changing internal fluid pressure. This may be described in terms of an effective stress change linked to poroelastic behaviour;  
• The poroelastic behaviour tends to be much greater at lower stresses;
• The poroelastic behaviour tends to be quite directional and therefore needs to be described by a tensor. The direction and magnitude of the poroelastic tensors correlate with the cleating in the coal;
• The use of uniaxial testing to determine the elastic properties of coals and other sedimentary rocks is generally quite inappropriate.

The consequences of these findings include:

• The simple linear elastic models used in most rock mechanics analysis of sedimentary rocks, including coals, are subject to a high degree of uncertainty, and may well be materially unrepresentative of actual conditions.
• The numerical models on which so much analysis is based have significant limitations. These need to be properly understood; they cannot necessarily be fixed by calibrating the model with field results if the models being used are fundamentally wrong.  
• Only those coals that exhibit fluid effects within the coal structure, including poroelasticity, may outburst. Such material could be a solid coal exhibiting poroelasticity to failure, or gouge material which is essentially plastic, depending on the rate of stress within it.
• The importance of confining stress on the stiffness of so many coals indicates that the cores of pillars may be far stiffer than the outsides and therefore the vertical stresses in their cores may be very high.