International Journal of Rock Mechanics and Mining Sciences
Numerical study of flow anisotropy within a single natural rock joint
Introduction
Understanding water flow through a rock mass is a fundamental issue in many areas of rock engineering, such as tunnel excavations, under the groundwater table, oil and gas reservoirs, underground nuclear waste disposal, or foundations in fractured rock masses. The flow through a discontinuous rock mass can be divided into flow through discontinuities and flow through the rock matrix. However, rock matrix permeability is often at least two orders of magnitude lower than rock joint permeability [1], [2]. It can, therefore, be assumed that the flow is governed only by discontinuities, and it is thus of prime importance to fully understand their hydromechanical behaviour to foresee how stress fields in situ can influence the fluid transport through the fracture.
Many investigations have been performed over the past three decades on hydraulic and mechanical properties of natural rock joints and on the influence of external applied normal and shear stress [2], [3], [4], [5], [6], [7], [8], [9], [10]. The roughness of the rock walls has early been identified as a key parameter of the mechanical behaviour of rock joints [4], [11]. Barton et al. [11] have proposed the JRC coefficient, an empirical parameter, to quantify the roughness but other approaches (e.g. geostatistical or fractal) have also been followed to quantify the roughness and its sensitivity to the scale [12], [13], [14], [15].
The joint roughness governs the mechanical response of rock discontinuities, either in terms of stresses or displacements, as well as its hydromechanical behaviour. Indeed, an increased (or decreased) void space due to dilation (or contraction) will lead to an augmentation (or reduction) of the hydraulic conductivity. The roughness evolves significantly during a shear test [16], [17] and several authors have studied the impact of the interface asperity degradation on the hydro-mechanical response [5], [17], [18], [19], [20], [21], [22] and on the alteration in hydraulic conductivity. If the fracture conductivity increases when rock wall asperities are worn off, it can also decrease when sheared rock particles close the flow path, which is known as the gouge material effect [22].
The flow through a rock fracture is governed by the Navier–Stokes equations, which are a set of three coupled non-linear equations difficult to solve. In case of a fracture bounded by smooth parallel walls, these former equations can be highly simplified and lead to the cubic law [23], [24], [25], which is still used in the literature in the rock joints context due to its simplicity (e.g., [26]) even if deviations from experimental data due to joint roughness have been observed [6], [27], [28]. Several attempts have been undertaken to improve the cubic law introducing roughness parameters [23], [29], [30], [31], [32] or reducing the value of the hydraulic aperture [19], [20], [21]. These corrections have not been that efficient and the main effect is a diminution of the total flow rate but the description of the flow anisotropy is poor. An equation more tractable than the Navier–Stokes equations and more accurate than the cubic law is the Reynolds equation, which is obtained by considerations of orders of magnitude [23], [33]. So far, this equation is considered to describe properly the flow anisotropy within a rock joint [17], [34].
This study intends to improve the description of flow anisotropy within a single discontinuity in an innovative way. The cubic law is combined to a reduction function taking into account the micro-scale roughness in order to define a reduced coefficient of permeability. This former is attributed to the elements constituting the rock joint, described as a porous medium in which Darcy's law is applied. The reduced coefficient of permeability at the micro-scale leads to a reorganization of flow towards most open channels. A good agreement between numerical results and experimental data after Hans [1] is thus obtained. In particular, flow anisotropy is well reproduced. A comparison with the Reynolds equation is given as well in order to highlight the efficiency of the model developed herein.
Section snippets
Experimental data
This study is based on the experimental data obtained by Hans [1] on the hydro-mechanical behaviour of rock joints under shearing. All experimental details are available in [1] and [35], especially the procedure to obtain the void map from the rock wall measurements. The chosen shear test (Test 1τ9) has been performed under a constant normal stress of 4 MPa and the relative tangential shear displacement, W, has been applied by successive steps of 2 mm until 10 mm. The anisotropy of the flow has
Numerical model
The void space between joint walls is modelled as a fully saturated porous medium, of given coefficient of permeability, subjected to a pressure gradient reproducing the experimental conditions [35]. In such a medium, Darcy's law governs the flow and the pressure field. The void space embedded between both joint walls (70 mm wide per 70 mm long in the initial configuration; see Fig. 3) is described using eight-node brick elements (C3D8P elements) [36], [37]. The total number of elements ranges
Calibration of the reduction function
The model developed herein is denoted MR (for micro-roughness). During the calibration phase, the coefficients of permeability of the classes are chosen in order to obtain flow rates as close as possible to the experimental values (see Fig. 9). Table 2 shows the average mechanical aperture for each class of the calibration model, the corresponding hydraulic aperture and the calibrated coefficient of permeability. Note that if the reduction in mechanical aperture ΔEel is greater than the value
Conclusions
The flow within a rock joint is governed by Navier–Stokes equations, which are quite difficult to solve. The simplest approximation of Navier–Stokes equation is the cubic law, which is known to not reproduce properly the flow distribution. The Reynolds equation is another approximation of the Navier–Stokes equations, which is easy to implement and considered, so far, to better reproduce the flow anisotropy. The model developed in this paper is based on the cubic law and on a reduction function
Acknowledgements
The authors thank Marc Boulon and Julien Hans of the Laboratory Soils, Solids, Structures of the University Joseph Fourier in Grenoble (France) for providing experimental data support. They also thank Kristian Krabbenhoft from the Centre for Geotechnical and Materials Modelling, University of Newcastle, for fruitful scientific discussions on finite elements.
References (45)
- et al.
Strength, deformation and permeability of rock joints
Int J Rock Mech Min Sci Geomech Abstr
(1985) - et al.
Water flow in a natural rock fracture as a function of stress and sample size
Int J Rock Mech Min Sci Geomech Abstr
(1985) - et al.
Single fracture under normal stress: the relation between fracture specific stiffness and fluid flow
Int J Rock Mech Min Sci
(2000) - et al.
Estimating joint roughness coefficient
Int J Rock Mech Min Sci
(1979) - et al.
Quantitative three-dimensional description of a rough surface and parameter evolution with shearing
Int J Rock Mech Min Sci
(2002) - et al.
Effect of shear displacement on the aperture and permeability of a rock fracture
Int J Rock Mech Min Sci
(1998) - et al.
Influence of rock joint degradation on hydraulic conductivity
Int J Rock Mech Min Sci Geomech Abstr
(1993) - et al.
An improved model for hydromechanical coupling during shearing of rock joints
Int J Rock Mech Min Sci
(2001) - et al.
Evolution of fracture permeability through fluid-rock reaction under hydrothermal conditions
Earth Planet Sci Lett
(2006) - et al.
An experimental investigation of hydraulic behaviour of fractures and joints in granitic rock
Int J Rock Mech Min Sci
(2000)
Experimental studies of the shear behaviour of rock joints
Int J Rock Mech Min Sci
Correlation between porosity, conductivity and permeability of sedimentary rocks—a ballistic deposition model
Physica A
A study of the quantitative relationship between permeability and pore size distribution of hardened cement pastes
Cement Concr Res
Permeability and microstructure of concrete: a review of modelling
Cement Concr Res
Review of a new shear strength criterion for rock joints
Eng Geol
Assessing the permeability characteristics of fractured rock
Geol Soc Am Spec Paper
Hydraulic characteristics of rough fractures in linear flow under normal and shear load
Rock Mech Rock Eng
Fluid percolation through single fractures
Geophys Res Lett
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