Numerically synthesized models of rough fractures in rocks are promising alternatives to the expensive profilometry process in different applications. The authors present a new powerful and flexible method targeted for the generation of high-quality synthetic fracture models. This method has been implemented in SynFracTM software, developed by authors. Statistical analysis of suites of synthetic fractures has been performed. As result of the analysis, the dependence of the mean fracture aperture on the main parameters of bounding surfaces has been obtained. This technique and the developed software are not restricted to use with rock surfaces, but can be applied for imaging and modelling of any rough surfaces in any material

Glover, P

Isakov, E

Ogilvie, S

White Rose Research Online

Image Anal Stereol 2001;20(Suppl 1):366-371Proc 8thECS and Image Analysis, September 4-7, 2001, Bordeaux, France 366USE OF SYNTHETIC FRACTURES IN THE ANALYSIS OFNATURAL FRACTURE APERTURESEVGENY ISAKOV, PAUL GLOVER AND STEVEN OGILVIEDepartment of Geology and Petroleum Geology, University of Aberdeen, Aberdeen, AB24 3UE,Scotland, U.K.e-mail: isakov@abdn.ac.uk, p.glover@abdn.ac.uk, s.ogilvie@abdn.ac.ukABSTRACTNumerically synthesized models of rough fractures in rocks are promising alternatives to theexpensive profilometry process in different applications. The authors present a new powerful andflexible method targeted for the generation of high-quality synthetic fracture models. This methodhas been implemented in SynFracTM software, developed by authors. Statistical analysis of suites ofsynthetic fractures has been performed. As result of the analysis, the dependence of the meanfracture aperture on the main parameters of bounding surfaces has been obtained. This techniqueand the developed software are not restricted to use with rock surfaces, but can be applied forimaging and modelling of any rough surfaces in any material.Keywords: numerical modelling, rough fractures.INTRODUCTIONComputational fluid flow modelling can be effectively applied to flows through rough fractures inrock. In order to do this the geometry of the fracture (2D or 3D) should be described numerically andthe flow boundary conditions should be stated. The topography of a real rock fracture can bepresented in the form of numerical description obtained using a profiling procedure. This procedure isusually expensive in terms of time, materials, labour and equipment required. An alternative way is tosimulate a numerical model of rough fracture, which has properties close to the real fracture. Brown(1995) developed a method of synthesis of numerical models of rough fracture. Synthetic profilesobtained with this method were analysed using statistical and spectral methods. The properties ofsynthetic profiles were found to be in reasonable agreement with data regarding natural fractures.Similar numerical model was used later by Glover et al. (1998a) in their work on the modelling offluid flows in rough fractures. Based on this model, we have developed a new method of numericalfracture synthesis. This method is more flexible and has several parameters of synthesis process to bevaried in order to achieve a high similarity of the simulated fracture compared to real rock fractures.The input parameters for the method can be obtained with an optical technology (see Ogilvie et al.this volume).High-quality synthetic numerical fracture models can be also used for investigations of thestatistical properties of rock fractures. Particularly, the mean fracture aperture dependence on theproperties of bounding surfaces can be obtained.  It is well known that fluid flow through roughfracture depends essentially on the fracture aperture. On the other hand, roughness of boundingsurfaces has also considerable influence on the fluid flow (Brown, 1987). However, the aperture andthe roughness are not entirely independent characteristics. In the frame of the present model both ofthem can be found depending on the model input parameters, which are related closely to theproperties of particular rock. For the aim of the present paper we assume that the input parameter canvary continuously, covering the range of discrete values, which correspond to various rocks.ISAKOV E ET AL: Synthetic fractures in the analysis of apertureProc 8thECS and Image Analysis, September 4-7, 2001, Bordeaux, France 367SYNTHESIS METHODAll three models listed above use the same fractal representation of rough fracture surfaces.Fractal surfaces are composed of a sum of Fourier harmonics, the intensities of which decay graduallyas a power of the wave-number, and with phases that are random. After the generation of the Fourierspectrum, an Inverse Fast Fourier Transform (IFFT) is applied in order to obtain the topography offracture surfaces. This method of generation of a fractal surface is called spectral synthesis (Saupe,1988).0.01.0Ιλ+Ιλ-R-R+Phase correlation, RHarmonic wavelength, λ23100.01.0R+R-Harmonic wave-number, kPhase correlation, R 132a bFig. 1. Correlation between phases of harmonic components vs. a: harmonic wavelength, b: harmonicwave-number. (1) Brown, 1995; (2) Glover et al., 1998; (3) New model.The two surfaces bounding the fracture are not exactly the same, and they are not completelyindependent. So the random phases of Fourier harmonics of these surfaces should be partiallycorrelated. Brown (1995) proposed that harmonics having wavelength greater than some mismatchlength λc  are perfectly correlated (the correlation coefficient, R = 1), while harmonics having shorterwavelength are independent ( R = 0 , Fig. 1a). Later Glover et al. (1998b) showed that thediscontinuity of the correlation function at the mismatch length λc  causes an unphysical behaviour ofsynthetic fractures produced with this method. They proposed a linear variation of the correlationcoefficient in the Fourier space within the interval from the wave-number k = 0  (infinite wavelength)to k c= 4π λ/  (wavelength λc / 2 , Fig. 1b, the corresponding non-linear dependence on thewavelengths is also shown in Fig. 1a). Glover et al. (1998b) also introduced the maximum fractionalmatching parameter R+ ≤ 1, which reflects the correlation between fracture surfaces at the largestwavelength used. In order to obtain correlated random phases, two independent random numbers ϕ1and ϕ 2 were mixedϕ ϕ ϕ3 1 21= + −R R( ) . (1)The resulting random number ϕ 3  and the initial random number ϕ1  are correlated with thecoefficient R . However, the random number ϕ 3  is not uniformly distributed (Fig. 2a), as it should be.For this reason, surfaces synthesized with this method were distorted at the corners (Fig. 2b).We have developed a new method of numerical synthesis of rough fractures, which is based onthe method of Glover et al., and avoids inaccuracies of the Brown and Glover et al. methods. Theproposed correlation function is shown in Fig. 1. Two new parameters were introduced in order toincrease the flexibility of the method. The first one is the minimum fractional matching parameter)0(≥−R , which reflects the correlation between surfaces at smallest wavelength used. The second is thelength of transition τ  from the minimum correlation scale λ−  to the maximum correlation scale λ+ISAKOV E ET AL: Synthetic fractures in the analysis of apertureProc 8thECS and Image Analysis, September 4-7, 2001, Bordeaux, France 368τ λ λ= −+ − ,   λ λλ τλ τ−=++ccc22( ) ,    τλλ += −+ . (2)The flexibility of new method allows the simulation of both the method of Brown( R− = 0, R+ = 1,τ = 0 ) and the method of Glover et al. ( R− = 0, τ → ∞ ).0 2π0P2πR 2π(1-R)Random phase value, ϕ3Probability densitya bFig. 2. Numerical model of rough fractures by Glover et al. (1998). a: Non-uniform probabilitydensity distribution of random phase 3ϕ . b: Sample of synthesized numerical fracture. Non-uniformdistribution of phases causes aperture increasing at the corners.Partially correlated random phases of Fourier harmonics were produced with a kind of MonteCarlo method. Two random sequences ϕ1  and ϕ 2  of a considerable length (10 104 6−  of randomnumbers) were generated. Then the sequence ϕ 2  was rearranged in a random way until the correlationbetween sequences reaches the desired value R . It is quite obvious that this procedure does not affectthe distribution of random value ϕ 2 , however it may introduce some autocorrelation to the sequenceϕ 2 . In order to avoid this effect, authors picked up random pairs from sequences (ϕ1 , ϕ 2 ) instead oftaking these pairs in order, one by one.We have implemented this new method of numerical synthesis of rough fracture model as asoftware algorithm (SynFracTM). This program is able to generate effectively sets of high-qualitysynthetic fracture models (Fig. 3) automatically or under manual control. SynFracTM software has agraphical user interface (GUI), which makes program easy to understand and use.As mentioned above, synthetic numerical fractures can be used in computational fluid flowmodelling. However, this is not the only application of synthetic fractures. As synthetic models have ahigh similarity to real fractures in rock, they may be used for collection of statistical data, when thereis a lack of real profiling data. In this work we use synthetic fractures to emphasize the relationshipbetween the properties of rock surfaces bounding the fracture and the mean fracture aperture. Indeed,this methodology and software can be applied whenever a rough surface or fracture is required in anymedium.  For example, this software has recently been used to provide large-scale (10 km ×  10 km)rough surfaces for use in satellite image modelling.ISAKOV E ET AL: Synthetic fractures in the analysis of apertureProc 8thECS and Image Analysis, September 4-7, 2001, Bordeaux, France 369a bFig. 3. Typical synthetic fracture models produced with SynFracTM software. a: Upper and lowerbounding surfaces of the fracture. b: Resulting aperture variation across the fracture. Note that thelack of long wavelength in the aperture. This is the result of the fracture surfaces being matched atthe larger wavelength.RESULTSA large number of synthetic rough fractures (100×100 mm) in rocks were created using SynFracas a function of fractal dimension (from 2 to 2.4), standard deviation of bounding surfaces (from 0.01to 5 mm), mismatch length (from 1 to 50 mm), and minimum and maximum matching fractions (from0 to 20% and from 80% to 100%, respectively). For each set of parameters a suite of 10-30 fractureswas created. Each fracture was analysed to ascertain whether the resulting synthetic fractures hadparameters, which matched the synthesizing parameters. In this way we verified the synthesizingalgorithms and their implementation in SynFrac  software.The mean arithmetic apertures of the resulting fractures were obtained for each suite of fractures,and have been examined as a function of (i) surface asperity distribution (standard deviation), (ii)fractal dimension, (iii) anisotropy, and (iv) mismatch parameters.Dependence of mean fracture aperture on the standard deviation of bounding surfaces is shown inFig. 4 (isotropic fractal dimension 2.2, mismatch length λ c = 10  mm, transition length τ = 20  mm).It is quite obvious that there is a strong proportionality relation between these values, as seen in Fig.4. The scattering in the mean fracture aperture increases as well as the mean fracture apertureincreases, so the relative scattering values are constant.As expected, the fracture aperture depends non-linearly upon the fractal dimension (Fig. 5). Twodata sets are presented in this figure, symbols ∆ and ∇  correspond to surface standard deviations of0.3 mm and 0.6 mm, respectively, and other parameters are the same as in Fig. 4. One can see that thechange of the standard deviation causes a proportional change of fracture aperture again. It appearsalso that the fracture aperture increases with fractal dimension as the roughness of fracture surfacesincreases.ISAKOV E ET AL: Synthetic fractures in the analysis of apertureProc 8thECS and Image Analysis, September 4-7, 2001, Bordeaux, France 370Standard deviation  (mm)of surfacesMean fracture aperture (mm)0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.51.01.5Mean fracture aperture (mm)Fractal dimension of surfaces (-)2.0 2.1 2.2 2.3 2.40.00.20.40.60.81.01.21.4Fig. 4. Mean synthetic fracture aperture as afunction of the standard deviation of surfaces.Fractal dimension 2.2, mismatch length 10 mm,transition length 20 mm.Fig. 5. Mean synthetic fracture aperture as afunction of the fractal dimension of surfaces.Mismatch length 10 mm, transition length 20mm, standard deviation of surfaces 0.3 mm (∆)and 0.6 mm (∇ ).It was found, that isotropic fractures have the least mean aperture, if all other parametersremaining constant (Fig. 6, fractal dimension 2.1, standard deviation 0.5 mm, other parameters are thesame as in Figs. 4 and 5). The mean aperture increases as anisotropy appears.It was mentioned above, that variation of the transition length parameter yields smooth transitionbetween Brown and Glover et al. methods. The variation of the mean fracture aperture during thistransition is shown in Fig. 7. The method of Brown gives the smallest values of fracture aperture andcan underestimate it. The mean aperture of the synthetic fracture increases as the transition lengthincreases up to 80 mm. As the transition length become comparable to the size of whole fracture (100mm), further increasing of the transition length does not affect the fracture properties. A largetransitional length causes considerable scattering of mean aperture values, because the correlationbetween long-wave harmonics with highest amplitude becomes random.0.1 1 100123Mean fracture aperture (mm)Anisotropy of surfaces (-)Mean fracture aperture (mm)Transition length,  (mm)τ1 10 100012Brown model Glover et al. modelFig. 6. Mean synthetic fracture aperture as afunction of the anisotropy of surfaces. Fractaldimension 2.1, mismatch length 10 mm, transitionlength 20 mm, standard deviation of surfaces0.5 mm.Fig. 7. Mean synthetic fracture aperture as afunction of the transition length. Fractal dimension2.2, mismatch length 10 mm, transition length 20mm, standard deviation of surfaces 0.5 mm.ISAKOV E ET AL: Synthetic fractures in the analysis of apertureProc 8thECS and Image Analysis, September 4-7, 2001, Bordeaux, France 371ACKNOWLEDGEMENTSThis work was sponsored by NERC as part of the Micro-Macro thematic program ongoing in the U.K.REFERENCESBrown SR (1987). Fluid flow through rock joints: the effect of surface roughness. J Geophys Res 92:1337-47.Brown SR (1995). Simple mathematical model of a rough fracture. J Geophys Res 100:5941-52.Glover PWJ, Matsuki K, Hikima R, Hayashi K (1998a). Fluid flow in synthetic rough fractures and application tothe Hachimantai geothermal HDR test site. J Geophys Res 103(B5):9621-35.Glover PWJ, Matsuki K, Hikima R, Hayashi K (1998b). Synthetic rough fractures in rocks. J Geophys Res103(B5):9609-20.Saupe D (1988). Algorithms for random  fractals. In: Peitgen H-O, Saupe D, eds. The science of fractal images.New York: Springer-Verlag, 71-136.

Use of synthetic fractures in the analysis of natural fracture apertures

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Use of synthetic fractures in the analysis of natural fracture apertures

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