Relationship between the geometrical and structural properties of layered fractured rocks and their effective permeability tensor. A simulation study
Introduction
Fluid flow modelling in fractured rocks is usually addressed by considering (i) discrete fracture models, (ii) equivalent continuum models using effective properties for discrete grids, or (iii) hybrid models that combine discrete large features and equivalent continuum grid properties (Lee et al., 2001).
In discrete fracture models (i), the matrix is generally assumed to be impermeable, which implies that the resulting permeability drops to zero when there is a loss in fracture connectivity (Bour and Davy, 1997, de Dreuzy et al., 2001). Because of their complexity, the use of discrete models is generally limited to small-scale problems (Jourde et al., 2002a).
The equivalent continuum models (ii) are widely used for field scale studies. Typically, these models include a limited number of regions in which physical properties (permeability or dispersivity) are assumed uniform. The physical properties computed from a fracture network are then averaged for the total rock volume, which includes fractures and matrix (Hsieh and Neuman, 1985, Long and Witherspoon, 1985, Jourde et al., 2002b).
Effective properties can be computed for a representative equivalent volume (REV) (Bear, 1972), in which the hydraulic properties change little with variation in rock volume. However, field measurements (Clauser, 1992, Cravero and Fidelibus, 1999, Schulze-Makuch et al., 1999), as well as numerical results (Margolin et al., 1998, Castaing et al., 2002, de Dreuzy et al., 2001) have shown that the permeability of a rock mass increases with its size. This scale effect has been explained as follows: when the volume of investigation increases, it either intersects more conductive discontinuities (morphological effect), or the relative arrangement of directional fracture sets changes (geometrical effect). These effects may lead to the absence of an REV at any scale (de Marsily, 1985, Pavlovic, 1998, Panda and Kulatilake, 1999).
Because of the scale dependency of geometrical and hydraulic properties, hybrid models (iii) combining discrete fracture and equivalent continuum models have been proposed (Clemo and Smith, 1989, Wen et al., 1991). The discrete part of the model describes the distribution of faults and the transfer within the “matrix”, whose equivalent permeability is computed for an REV of the fractured rock mass compartmentalized between faults.
This later approach implies a detailed description of the rock mass, whose effective permeability properties can be addressed according to two different approaches: (i) either one uses analytical expressions while considering the statistical and geometrical properties of the discontinuity network (Snow, 1969, Sagar and Runchal, 1982, Oda, 1985, Mania et al., 1998, Doolin and Mauldon, 2001). This formalism is well suited for networks of simple geometry, but is inefficient when the rock mass is characterized by a complex architecture of discontinuities (Paillet, 1985, Massonat, 1998, de Dreuzy et al., 2001, Castaing et al., 2002); (ii) Or one proceeds to steady state flow modelling throughout the rock mass, while considering various directions of flow depending on the given hydraulic gradient. The effective permeability is then determined in analogy with Darcy’s law (Long et al., 1982, Long and Witherspoon, 1985, Long and Billaux, 1987, Massonat and Manisse, 1994). This approach is well suited for the study of various and complex natural fractured networks.
In this study, we used the both aforementioned approaches to determine the equivalent permeability properties of a layered fractured rock mass characterized by one set of bedding perpendicular joints and one set of bedding parallel joints.
This type of fracture network is characterized by a wide range of spatial relationships between bedding perpendicular and bedding parallel joints, which depend on the properties and thickness of the bedding surface infilling, and on the stress states during genesis and propagation of fractures. As a consequence, the persistence of bedding perpendicular joints across bedding planes varies highly, and this can be characterized statistically (Ladeira and Price, 1981, Helgelson and Aydin, 1991, Rives et al., 1992, Petit et al., 1994). This persistence is an important parameter for the permeability properties normal to bedding surfaces. In a similar way, the characteristics of geometrical (spacing and length distribution) and morphological (hydraulic aperture) parameters of bedding perpendicular joints should constrain the permeability normal to bedding.
The role of the aperture on the permeability normal to bedding and its consequences on the scale of an REV was studied in 2D by Doolin and Mauldon (2001). These authors pointed out that a longer chain has a greater chance of containing a weak link than a shorter chain. Similarly, since flow in the bed normal direction is controlled by the least permeable plane or bedding plane, adding layers increases the chance of an impermeable bed or bedding plane. The scale of homogeneity of the fracture system may not imply the scale of an REV. For a simple fracture geometry resembling courses of brick work, the scale of homogeneity of the fracture system bounds a single block of rock (brick), but the permeability drops rapidly as the number of layers increases (Doolin and Mauldon, 2001). These authors showed that, if an effective medium is assumed where the permeability changes very little with an increase in the number of layers, an order of magnitude difference between the scale of fracturing and the scale of an REV can be demonstrated.
In this 3D study, there is no attempt to investigate whether the samples used are on a REV scale. The work is made at the scale of homogeneity of the fracture system (Doolin and Mauldon, 2001) which bounds a single block of rock (brick), that we referred to as an elementary volume.
As we wanted to focus essentially on the effect of geometrical and structural properties on the effective permeability tensor of a 3D block, we considered the same equivalent hydraulic aperture for the bedding perpendicular joints than for the bedding planes joints.
We first present the three-dimensional structural model (REZO3D) and discuss the variation range of geometrical and structural parameters encountered in rock masses made of one set of bedding perpendicular joints and one set of bedding parallel joints. Secondly, we expose the methodology for determining the effective permeability tensor components and analyse their sensitivity to the geometrical and structural properties of the synthetic network. The implications for field characterization of effective permeability properties of a rock mass are then discussed. Finally, we compare the results inferred from the three-dimensional model with permeability values obtained with a simple analytical model. The pertinence of this later model for estimating a 3D effective permeability tensor is then discussed.
Section snippets
Genesis of the discontinuity networks
The three-dimensional model (REZO3D) allows the simulation of networks made of two orthogonal generations of bedding perpendicular joints in a tabular stratified medium. In this study, we consider networks made of one set of bedding perpendicular joints and one set of bedding parallel joints (bedding planes), which are continuous planes of infinite extent (Fig. 1). The later assumption is supported by field observation of sedimentary rocks.
Most fracture systems where one set is dominant have in
Results and discussion
For each synthetic network, we numerically determine (as an output of the structural model REZO3D) the density Dfc (m2/m3) of bedding perpendicular joints crosscutting the elementary volume (cf. Fig. 4b). Given the assumed number of strata, the bedding planes density within the rock mass of unit volume remains Dbp = 4 m2/m3 for all the networks.
An analytical expression accounting for these two parameters (Dfc and Dbp), the equivalent hydraulic aperture ah (0.08 mm) and the corresponding
Conclusion
We used a three-dimensional numerical model to generate a wide range of network geometries with realistic crosscutting relationships between bedding perpendicular joints and bedding parallel joints. The effective permeability was considered to be the contributions of both bedding perpendicular joints and bedding parallel joints which we assumed to have the same hydraulic apertures. The geometrical and structural properties of the networks were determined, which allowed estimating the influence
Acknowledgements
The authors wish to thank Gh. de Marsily, an anonymous reviewer, and the editor for their helpful comments and suggestions, that greatly helped to improve the present contribution.
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2017, Advances in Water ResourcesCitation Excerpt :In this work, a genetic code for discrete fracture network (DFN) generation (Jourde 1999; Josnin et al., 2002) has been employed and further developed, which can capture realistic spatial distribution of multiple joint sets in layered systems. Extensive studies have been done to investigate fluid flow though such a complex discontinuity system in the bed-parallel direction (Fig. 1a; e.g. Taylor et al., 1999; Bai et al., 2000a; Matthäi and Belayneh, 2004; Belayneh et al., 2006; Jourde et al., 2007). However, only a few studies have considered fluid flow in the bed-normal direction of the layered system (Fig. 1b; Doolin and Mauldon, 1996; Morin et al., 2007; Jourde et al., 2007).
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