Description:

_Journal of Structural Geology 32 (2010) 878–885_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _Determination of geometric parameters of fracture networks using 1D data_ _Tivadar M. Toth* _Department of Mineralogy, Geochemistry and Petrology, University of Szeged, P.O. Box 651, H-6701 Szeged, Hungary_ _article info_ _Article history: Received 25 February 2008 Received in revised form 13 April 2009 Accepted 19 April 2009 Available online 3 May 2009_ _Keywords: Fracture network simulation Scanline Length exponent Fractal geometry_ _abstract_ _To simulate a suitable fracture network for hydrogeological modelling, input statistical data of the individual faults as well as fracture sets should be determined first using either 2D sections or 1D scanlines. Although the accuracy of this measurement is fundamental, exact determination is rather problematic and is usually possible only at a particular scale. This paper introduces a coupled method for computing length exponent (E) and spatial density (Dc), the two most essential parameters for modelling fracture networks. To calculate the length exponent, data sets of at least two independent imaging processes are needed. Utilizing different sensitivity thresholds of the two methods and the well-known analytical form of a fracture length distribution function, its parameters can be calculated. To estimate the spatial density of fracture centres in 3D, the series of intersections should be analysed as a fractional Brownian motion and then calibrated with virtual wells simulated with optional modelling software. The method makes fracture intensity logging possible along scanlines. Based on these approaches, there is no need to import fracture parameters from the outcrop survey or from other parts of the reservoir because all geometric information of the fracture system refers to the rock body under examination. Using site-specific parameters makes fracture network modelling more reliable._ _? 2009 Elsevier Ltd. All rights reserved._ _1. Introduction_ _Under appropriate physical conditions, rock deformation produces brittle structures. Since the resulting fracture system has an essential role in the hydraulic behaviour of the rock body, reconstructions of both the structural evolution and the spatial appearance of the fracture network are crucial. Faults regularly appear at different scales, from the submicroscopic size to the kilometre scale (Alle`gre et al., 1982; Turcotte, 1992; Ouillon et al., 1996) with structural and geometric features comparable at all magnitudes (Korvin, 1992; Turcotte, 1992; Long, 1996; Weiss, 2001). Such behaviour ensures the theoretical background of fracture network modelling concept, in which structural data measured at micro- or meso-scales are used to upscale information to larger dimensions. Mathematical simulation of a 3D fracture network at the reservoir scale, taking into account also the lithological and structural setting, is especially important because hydraulically active fracture sets are usually out of the scale of both microstructural and seismic measurements (Paillet et al., 1993; Childs et al., 1997)._ _Simulation generally has two sequential steps. At the first point, geometric parameters of the individual faults as well as_ _* Tel.: ?36 62 544640. fax: ?36 62 426459. E-mail address: mtoth@geo.u-szeged.hu_ _0191-8141 $ – see front matter ? 2009 Elsevier Ltd. All rights reserved. doi:10.1016 j.jsg.2009.04.006_ _fracture sets must be determined. These parameters should serve the firm base for the simulation itself using appropriate modelling software. In the last decades, numerous algorithms and programs have been developed (for example Long (1996), Zhang and Sanderson (2002), FracMan (Dershowitz et al., 1993), FracNet (Gringarten, 1998), RepSim (Toth et al., 2004) etc.) to solve diverse problems regarding fractured reservoirs. Whilst the exact spatial definition of hydraulically active faults is desirable, this is not generally possible and so in most cases a simulated network is used for hydrogeological assessment. That is why most approaches aim to generate a stochastic reconstruction of a network of individual fractures (discrete fracture networkdDFN methods). To do so, in addition to structural parameters (e.g. fault generations, kinematic indicators, fracture filling mineral paragenesis) additional geometric parameters of the fault set such as length, aperture, orientation and spatial density are required. Although accuracy of determination of these parameters is fundamental to the success of modelling, measurements are rather problematic (Yang et al., 2004) and are usually possible exclusively at a particular scale (La Pointe and Hermanson, 2001; Zimmermann et al., 2003). Besides using outcrops, geological cross-sections, borecores or thin sections for measurement on 2D sections, parameter estimation is mainly possible along 1D scanlines (Priest and Hudson, 1981; La Pointe and Hermanson, 2001; Priest, 2004 and references therein), supplied by well-logs._ _T.M. Toth Journal of Structural Geology 32 (2010) 878–885_ _879_ _The aim of the present paper is to introduce novel algorithms for determining fracture length distribution and spatial density using 1D datasets. Since density can be defined in several different ways (e.g. Long, 1996), the estimation methods should be harmonized with the modelling system as well._ _2. Geometric parameters of fracture systems_ _In addition to structural data, quantitative parameters also play an essential role in describing fracture systems. A single fracture is usually a finite, complexly buckled 2D surface that can be approximated by planes (Chiles and de Marsily, 1993). In a homogeneous, isotropic rock body under pure tensile stress, the shape of a fracture is close to circular (Twiss and Moores, 1992); whereas in the case of stratified sedimentary rocks, a more anisotropic appearance (ellipse) is more common. However, the shape of fracture planes may be extremely complex due to overlapping structural effects; in order to make the simulated fracture network appropriate for hydrodynamic modelling, some limitations must be used. Since flow between smooth, parallel plates is the only fracture geometry that is amenable to exact treatment for hydrodynamic modelling (Witherspoon et al., 1980; Neuzil and Tracy, 1981; Zimmerman and Bodvarsson, 1996), fractures should be approximated by thin discs. And while the most regularly followed choice for fracture shape under these constraints is a circle (''penny-shaped''), other approximation estimates also occur. In what follows, fractures are modelled by penny-shaped cracks by taking into account that many other simulators (e.g. FracMan, Dershowitz et al., 1993) use synthetic faults of a polygonal shape. A circle in 3D is explicitly defined through the coordinates of the centre, the radius and orientation. In the case of fault systems, the spatial function of the geometric centres as well as the distribution functions of length, strike and dip must be determined. Because hydraulic characterization of fracture networks presumes a positive volume of each fracture, circles are usually replaced by discs with a certain aperture (''parallel plate model'', Witherspoon et al., 1980). Symbols that will be used through the modelling process are collected in Table 1._ _One of the most important parameters concerning fluid migration through fracture networks is the size (diameter) of the fractures. Davy (1993), Bour and Davy (1997), de Dreuzy et al. (2001), Bonnet et al. (2001), among many others, inferred an asymmetric length distribution of fracture diameter, which, according to the most widely used model (e.g. Yielding et al., 1992; Min et al., 2004) follows the_ _Table 1 Symbols used in the text._ _Variable | Explanation of the variable ---|--- L | Diameter of a penny-shaped fracture D, D1, D2, D3 | Fractal dimension in general / Fractal dimensions of the fracture network traces in 1D, 2D, 3D Euclidean spaces / Fractal dimension of fracture centres of a 2D network trace / Fractal dimension of fracture centres in 3D Dc2, Dc3 | Parameters of the aperture function E, F | Parameters of the length distribution function Ea, Eb | Detection length thresholds of different fracture identification methods Da, Db | Detection aperture thresholds of different fracture identification methods H | Dip of a fracture F | Strike of a fracture w | Size of a box in the box-counting algorithm f(n) | Hurst exponent L1, L2 | Series of fracture intersection depths w | Length of an interval used for the R S analysis _power law function, where N(L) is the number of fractures with a diameter of L; E and F are the parameters of the fracture length distribution function._ _Aperture becomes essential only if the simulated fracture network is evaluated hydrogeologically, i.e. it is used for estimating porosity and permeability tensor. Exact determination of aperture is quite problematic (Vermilye and Scholz, 1995). The original aperture can be significantly modified due to water-rock interaction processes, either solution or precipitation. Aperture distribution also depends considerably on the orientation of the current in situ stress field (Allen and Roberts, 1982), which makes its measurement rather uncertain. Moreover, fluid migration in a crack is also a function of the roughness of the fracture wall (Kumar and Bodvarsson, 1990; Kumar et al., 1991; Liu, 2005), which further complicates the estimation of the effective aperture values. To solve this problem, Leckenby et al. (2005) suggest measuring the thickness of entirely cemented fractures as ‘‘paleo-apertures’’; whereas Keller (1998) uses computer tomography for the same purpose. Even if the measurement is uncertain, aperture, like fracture length, seems to follow a power law distribution function (de Dreuzy et al., 2002; Ortega et al., 2006). Furthermore, between these two parameters, a tight linear correlation can be assumed._ Ключевые слова: fracture intensity, loiseau, european, sensitivity, korvin, international, fractal analysis, fracture aperture, dimension, water, simulated, geometric, min, based, outcrop survey, density, hurst, la pointe, scaling, rock body, strike, fractured reservoir, centre, exact, bodvarsson, cimino, model, diverse, well-log data, scaling property, water resources, fractal dimension, resources, mo?, spatial, szeged, journal structural, mineralogy, limit, law, box, structural evolution, determined, elsevier, turcotte, number, wall, algorithm, hurst exponent, function, paper introduces, fracture, gudmundsson, imaging, intersection, geophysical, contaminant transport, mandelbrot, hand-collected specimen, linear function, gy, liu, kaszai, set, sensitivity threshold, fault, essential parameter, size, depth interval, matsumoto, measurement, length, mechanics, structural geology, shape, depth, roberts, fracture seed, fractals, fracture length, fracture diameter, laubach, rock, detection limit, trace, crystalline rock, bulletin, kranz, modelling, data, smaller cube, geophysical letters, wa, to?th, geological society, davy, geology, parameter, log, zimmermann, geophysics, ra? gy, signi?cantly, numerical experiment, method, process, fracture network, fracman, mo? ra?, approach, dip, point involved, bean, analysis, ra?, eds, press, journal structural geology, exponent, series, estimate, long, nature, simulation, eds flow, witherspoon, determination, individual fault, priest, hydraulic conductivity, essential, aim, fractal geometry, sanderson, reservoir, spatial density, geothermal, gringarten, hydraulic, linear, rock mechanics, core imaging, acta, fracture centre, granite, gudmundsson vermilye, threshold, hydraulic property, geosciences, point process, network, hungarian, body, pointe, power, uncertain, telesca, scanlines, hungary, length distribution, evaluation, aperture, distribution, coupled method, power law, hydrology, bhtv, hermanson, keller, fractal, point, barton, core, calculated, uid migration, letters, geological, la, to?, calculation, length exponent, journal, detection, hirata, essential role, structural, scale, trace length, fractured, journal geophysical, geological institute, case, to ro, standard deviation, granite wall, geometric parameter, interval, cube, geometry