Synthetic rough fractures in rocks

Transcription

1 _ JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. B5, PAGES , MAY 10, 1998 Synthetic rough fractures in rocks P. W. J. Glover Institute of Fluid Science, Tohoku University, Sendai, Japan K. Matsuki and R. Hikima Department of Geosciences and Technology, Tohoku University, Sendai, Japan K. Hayashi Institute of Fluid Science, Tohoku University, Sendai, Japan Abstract. Natural fracture serve a very important function in the transport of fluid through rocks, as well as in the flow of electrical charge and heat. The creation of numerically synthesized fractures can aid the study of the physical processes connected with fractures. Synthetic fracture is the term used to describe fractures that are created numerically in such a way that they share the same mean geometrical characteristics as specific natural fractures measured by profiling, by a process known as tuning. We have modified methods for producing synthetic rough surfaces whose geometric properties are tuned to mimic natural fracture surfaces in rocks in order to create synthetic fractures that are statistically identical to those found in rocks. One important such modification has been the incorporation of a method that allows the surfaces to be matched at long wavelengths and unmatched short wavelengths, with the degree of matching varying smoothly in between, as it does for real fractures. We have compared numerically synthetic fractures created using the new method with fractures created using an existing technique that uses a mismatch wavelength as a sudden discontinuity between matched and unmatched behavior, as well as data from fractures in real rocks. This comparison has shown that the new technique provides much more realistic numerically synthesized fractures than previous methods. Synthetic fractures created with the new method have been used in normal closure and fluid flow modeling, and the results are reported in a companion paper. 1. Introduction dependent upon the presence and geometrical properties of fractures [Brown and Scholz, 1985, 1986]. Microscopic mod- There now is increasing awareness of the great effect fracels of two contacting rough surfaces have been used to derive tures have on the mechanical and transport properties of rocks. the elastic properties of single fractures including both normal Such fractures may range in scale from extremely small miand shear stiffness [Yoshioka and Scholz, 1989a, b], and to crocracks, with a characteristic length scale smaller than the predict friction, wear, and the stability of shear behavior in grain size of the rock, up to large fault systems. It is known rock joints [Swan and Zongqi, 1985]. In all these models a that many of the statistical geometrical characteristics of natucontrolling influence is exercised by the interaction between ral and hydraulically induced fractures in rocks are fractal. For asperities on each surface, described by the aperture (or the example, the length, aperture, and volume distributions of composite topography) of the two interacting surfaces. The fractures in crystalline rocks can almost always be described shape, size, and number of asperity contacts and the local using fractal concepts [e.g., Hiram, 1989; Main et al., 1990; slope of the surfaces are particularly important parameters in Sammonds et al., 1994; Watanabe and Takahashi, 1995]. such models. Additionally, the surface of a fracture has a roughness that is Clearly, the transport properties of fractured rocks are also also possible to describe in terms of fractal geometry [e.g., controlled by the size, shape, and connectivity of the fractures A vnir et al., 1984; Katz and Thompson, 1985; Thompson et they contain. As expected, the capacity of rock to conduct fluid al., 1987; Hansen and Skjeltorp, 1988; Krohn, 1988; or electrical charge depends upon the distribution of apertures Thompson, 1991; Schmittbuhl et al., 1995; Matsuki et al., in the rock, and some term indicating the extent to which such 1995; Aharonov and Rothman, 1996]. "fractures" are connected. When rocks are subjected to pres- The mechanical properties of rocks, described by their bulk sures, temperatures, and stresses that are commonly found in elastic constants and shear strength, are known to be strongly Earth's upper crust, fractures either close or propagate. During Now at Department of Geology and Petroleum Geology, Univer- closure, it is the interaction between asperities on the surfaces sity of Aberdeen, King' s College, Aberdeen, Scotland. that has the greatest effect, either propping open cracks that would otherwise close under the influence of an overburden Copyright 1998 by the American Geophysical Union. Paper number 97JB /98/97JB pressure or blocking and increasing the tortuosity of fluid flow and electrical conduction paths [Gavrilenko and Gudguen, 1989; Durham and Bonner, 1994]. Thus the transport proper-

2 9610 GLOVER ET AL.: SYNTHETIC ROUGH FRACTURES IN ROCKS ties of a fractured rock depend upon the aperture distribution and the fracture connectivity. These are, in turn, related to the mechanical properties of the rock, which are modified by the presence of rough fracture surfaces. Fluid flow may also be affected by perturbation of laminar flow caused by the fracture surfaces. This is because there will always be some critical crack closure at which the flow rate and aperture distribution give rise to turbulence, even though geological fluid flow rates in a rock are rarely sufficiento perturb laminar flow in open macroscopic fractures. The characterisation of the effects of rough fracture surfaces has been reported by Brown et al. [1986], Brown [1987a, b], Pyrak-Nolte [1988], Zimmerman et al. [1991], Brown [1995], Matsuki et al. [1995, 1996], and 1994] and in many papers [e.g., Mandelbrot, 1983; Brown and Scholz, 1985; Brown, 1987a, b; Wang et al., 1988; Power and Tullis, 1991; Brown, 1995; Schmittbuhl et al., 1995; Matsuki et al., 1995, 1996]. It is not our intention to provide a full literature review here. However, we will briefly review some of the key concepts in the analysis of natural, and generation of synthetic fractally rough fracture surfaces and fractures. It is best to begin with a definition of a fractal fracture in rock. By fractal fracture, we mean a fracture occupying threedimensional space with two surfaces, each with a fractal dimension between 2 and 3. In general, a statistical description of either of the surfaces that goes to make the fractal fracture is given by specifying two basic functions. Kojima et al. [ 1995]. 1. The probability density function (PDF) for surface It is important for us to study the way the geometry of fracture surfaces affect the mechanical, hydraulic, electrical, and thermal conductivity properties of the fractured rock mass. One approach to this problem is to use numerically synthesized fractures. This approach has the great advantage that it allows the various geometrical parameters that describe numerically synthesized fractures to be varied independently in order to discover the effect of each. It also has the constraint that the method for creating numerical synthetic fractures should mimic very well natural fractures in rocks for the synthetic study to be representative of natural fractures. Until now, numerical synthetic fractures have been created from unmatched fractal surfaces [Amadei and lllangsekare, 1994]. More recently Brown [1995] has reported a simple code to create realistic synthetic rock fractures. This code is based upon that of Saupe [1988] but includes simple single cutoff mismatching that allows the two surfaces that make up the fracture to be perfectly matched at long wavelengths (above the mismatch wavelength), and behave independently at shorter wavelengths, as observed in natural rock fractures [Wang and Narasimhan, 1985; Brown et al., 1986; Brown, We will first briefly describe the mathematical foundation of the geometrical parameters that describe fractures with rough surfaces including an extended discussion of mismatch length scales, which is an extremely important concept in the production of realistic numerical synthetic fractures. We will heights above some mean datum describes the distribution of surface heights irrespective of the horizontal spatial position. Surface height distributions can only be compared for truly stationary data, i.e., data which have a well-defined mean. Truly fractal surfaces are nonstationary and do not have a well-defined mean. This has implications for the method for numerically generating partially matched fractal surfaces that are discussed later in this paper. An example of a profile across a fractal surface is given in Figure 1, together with its PDF and cumulative probability function. 2. The power density spectrum (PSD) describes the spatial correlation (or texture) of heights on the surface, in which the fractal nature of the surface resides. Figure 2 shows an example of such a plot for a typical surface and a typical aperture resulting from two similar such surfaces in a siliceous tuff [Glover et al., this issue]. Here the value max represents the lowest wavelength present in the analyzed data and is related to the length of the profile, and min represents the highest frequency information in the data set (twice the sampling step). The value res represents twice the shortest wavelength at which the profiling instrument reaches its vertical resolution. Data at shorter wavelengths than,re Can be recognized by the apparent decrease in the gradient of the power spectrum. The range of values indicated by,. representhe approximate range where mismatch wavelength of various definitions [Brown, 1995] occurs, and corresponds to where the power spectrum of the aperture deviates from that of each of the surthen very briefly describe the numerical generation of synthetic faces. fractures with rough surfaces in detail. The new technique used to control mismatching will be described in detail, as it differs from that used by Brown [ 1995] by allowing the extent of mismatching to develop gradually with wavelength. A comparison of the effectiveness of the new technique with that of Brown [ 1995] and data obtained from natural rock fractures Glover et al. [this issue] are described subsequently. The tuning of such fractures with experimentally derived data allows synthetic fractures to be produced that are extremely good analogs for real fractures. This allows tuned synthetic fractures to take the place of real data in mechanical, electrical, and hydraulic numerical modeling, allowing the effect of varying single geometric parameters defining the surface to be studied easily. A normal closure and fluid flow case study using the new technique to generate synthetic fractures is given in a companion paper Glover et al. [this issue]. It is often assumed that surfaces have a Gaussian distribution of surface heights. However, the experimental reality is often different, especially if the rock contains a significant proportion of fine cementing material between grains [Glover et al., this issue]. If a Gaussian distribution is assumed, then a full statistical description of a single stationary twodimensional isotropic rough surface is provided by (1) the standard deviation of the Gaussian distribution, (2) the fractal dimension, and (3) the upper and lower length scales between which fractal scaling occurs. It is important here to distinguish between stationary surfaces and nonstationary surfaces. Surface heights are only well-defined for stationary surfaces, which implies that the surface has a well-defined mean. Truly fractal surfaces do not have a well-defined mean, and this property is critical to many of the applications of fractal description. The consequence of nonstationarity is that it is possible that no surface height distribution in the standard sense may be able to describe the surface [Power and Tullis, 1991]. 2. Mathematical Description of Fracture Surfaces In this work we use fractal surfaces; however, the surfaces and The mathematical description of rough surfaces is well reviewed both in books [e.g., Peitgen and Saupe, 1988; Russ, derived apertures are not nonstationary for two reasons. First, the surfaces used in this work are matched at long wave-

3 GLOVER ET AL.' SYNTHETIC ROUGH FRACTURES IN ROCKS E 0.4 E.,..r 0.2 Az i I ( o Arbitrary Distance in Single Dimension [--Fractal Profile, D=1.251 Probability Density Function, p(z) Cumulative Probability Function, P(z) Figure 1. The physical manifestation of the probability function and probability density function when used to describe the distribution of surface heights of a profile across a fractal surface; after Brown [1995], and using a profile from a surface created using the algorithms described in this paper. lengths. This enables us to define a well-controlled and meaningful mean or average for the aperture. Second, the spectral synthesis method which we have used to create the synthetic fractal surfaces implicitly defines a zero mean for the finite fractal surface that is created. A well-defined mean can be taken for such finite synthetic fracture surfaces; however this mean would not be valid if the surface extended to infinity in any direction. We have used a Gaussian surface height distribution in this work; however, if the surfaces of the natural fracture have a different distribution of surface heights (such as lognormal), then a complete statistical description requires more definition than the three-point list given directly above [Ripley, 1981; Matsuki et al., 1996]. Additionally, if the surface has an anisotropic distribution of surface heights or fractal dimension, then this must also be taken into account. loo i Ap ' rofile E O.Ol u) 1E-6 o m E-8 1E Mismatch Length Scales In practice, if a fractal fracture created from two surfaces is to be similar to those found in real rocks, we must add a third parameter called the mismatch length scale [Brown, 1995]. This parameter accounts for the way that two mating surfaces in real rocks are matched at long wavelengths but behave independently of each other at short wavelengths [Wang and Narasimhan, 1985; Brown et al., 1986; Brown, 1995]. Failure to include a parameter to describe this behavior leads to the creation of numerically synthesized fractures, which are not similar to fractures in rocks. Experimental observations by Matsuki et al. [1995] and Brown [1995] show that natural fractures are correlated to some degree (matched) at long wavelengths, defined by the gross geometry of the original fracture path. However, at short wavelengths the surfaces are not identical. At these short wavelengths the surfaces behave independently, even if they retain some statistical similarity, and even share the same fractal dimension. These two different behaviors have been described mathematically by a critical wavelength,., called the mismatch length scale [Brown, 1995]. Above the mis- Spatial Frequency, l/ram match length scale the surfaces are correlated to some degree, Fibre 2. T icfl power sp fl density plot for one surface om and below the mismatch length scale they behave independa natural acture(the o er is almost identical) and the a re reently. sulting om the two acmre surfaces. e dam were obt n a er The critical length scale is a particularly difficult parameter the analysis of profi ng data in two pe endicul d fions across a acture in Hac mant si c us tuff. Labds are to define accurately in natural rock fractures because the is the laterfl extent of the profile, is the lateral resolution of the transition from matched surfaces to independent surfaces is profile (twi the lateral samp ng step),. shows the range where gradual, and one must take an arbitrary threshold at which the smatch wavelen hs of v ous deflations ur, d is fl e mismatch length scale is defined. One method of obtaining the wavelength at w ch the venicfl resolution of the profilometer is mismatch length scale is to plot the ratio of the power spectral reach. densities of the aperture to that of the surfaces, as a function of

4 9612 GLOVER ET AL.' SYNTHETIC ROUGH FRACTURES IN ROCKS 10 ing occurs at wavelengths greater than or equal to the average grain size of the rock; however, a much larger data set is needed to test this hypothesis. 1 We have found in our initial modeling that the use of 3,,. (2) (the smaller of the two length scales) leads to more realistic numerically synthesized fractures, when the power spectral densities of the surfaces and apertures of the resulting synthetic fractures are compared with those of profiled fractures in natural rocks. The method of comparison is discussed in full 1/) c m 1/) ½ (2) below. Subsequent use of our new technique for controlling the degree of mismatch between the two surfaces composing a synthetic fracture, which varies 3,,. until the best 1:1 correlation I is found between the PSD of the aperture of the synthetic Spatial Frequency, 1/mm fracture and the PSD of the aperture from profiling carded out Figure 3. Typical power spectral density ratio ( (30) plot for the on a natural rock fracture, results in values of 3,,. that are very ratio of the mean power spectral density of both surfaces of a natu- close to the empirically estimated value of 3,?, which is also ral fracture to that of the aperture resulting from the two fracture the length scale closesto the average grain size of the rock. surfaces, for the same fracture data as given in Figure 2. The labels Unfortunately, the simple application of such definitions of indicate the spatial frequency representing the two mismatch threshold mismatch values in the creation of synthetic fracwavelengths, 3,? and 3,?, described by Brown [1995]. tures leads to unrealistic results, particularly the underestimation of the standard deviation and mean of the aperture. Since these parameters are likely to have a first-order effect on the spatial frequency. We have denoted this ratio as (3,). Figure 3 efficiency of fluid flow in the fracture, we considered that it gives an example of such a curve for a natural fracture in a si- was necessary to improve the method of controlling the degree liceous tuff, where each of the surfaces has approximately the of matching between the two surfaces making up the synthetic same fractal dimension. Note that at large wavelengths the fracture. Below, we introduce a method of allowing the degree value of (3,) is much less than unity, indicating that the aper- of matching between two surfaces to vary smoothly as spatial ture is varying much less than the individual surfaces from wavelength varies. which it is derived. That is, the individual Fourier component amplitude for a particular spatial frequency for the aperture is much less than the corresponding Fourier amplitude for each of the surfaces that compose the fracture. This situation is only possible if there is a significant correlation between the two surfaces. Perfect matching at the largest wavelengths would be represented by =0; however, it is unlikely that this is ever 4. Fracture Modeling The basic approach using the spectral synthesis method is well covered both in books [e.g., Peitgen and Saupe, 1988; Russ, 1994] and in many papers [e.g., Brown, 1995]. In this short section we review some of the basic requirements of the achieved in natural physical systems. At small wavelengths, spectral synthesis method for numerically synthesizing frac- (3,) tends toward 2 in this example. This results from the variation of the aperture being equal to the sum of the variations of the surfaces heights of each surface. If the variations of the two surfaces are the same (as in this example), and the surfaces are independent, the variation of the aperture is equal to twice the variation of either surface. tals, and describe the improvements we have incorporated into the technique. The spectral synthesis method depends on defining a twodimensional field in frequency or wavenumber (k) space that contains complex Fourier components S(k) obeying a scaling law related to a given fractal dimension D [Brown, 1995]. When such a field is submitted to a two-dimensional inverse One definition of the mismatch length scale is then the spatial frequency at which the resulting curve crosses the =1 line, denoted 3,. ø) by Brown [1995] (parentheses have been inserted around the superscript here to avoid confusion with the mathematical operation). Another definition is the spatial frequency at which the resulting curve departs significantly from S(k)o k ', where the fractal dimension, D=(7-tx)/2 [Brown, the =2 line, denoted 3,,. (2) also by Brown [1995]. By defini- 1995]. Measurements of natural fracture surfaces [Brown and tion, the value of 3,. ( ) is always greater than that of 3,?. Scholz, 1985; Power et al., 1987; Power and Tullis, 1991, Both of these experimentally derived length scales have been calculated for a range of natural rock fractures by Brown [ 1995], and were found to lie between mm and 7.52 mm for ).(]), and between 0.18 mm and 3.3 mm for 37 ), where the profiles were between 13 mm and 52 mm long. We have calculated the mismatch length scales for 20 profiles in a tuffaceous siltstone, each of which was 42 mm long, and found these mismatch length scales to lie between 0.5 mm and 2.50 mm for ).o), and between 0.25 mm and 0.83 mm for 3,?. In the case of the rock studied in this work, the average grain size is 0.22 mm, although a great deal of the rock is composed of cryptocrystalline quartz and isolated large vitrous silica grains up to 4 mm in diameter. We hypothesize that fracture match- Fourier transform, the real part of the two-dimensional output is a fractal surface with the required fractal dimension [Saupe, 1988]. The data field containing the Fourier components must be symmetric in the two lateral dimensions, and take the form 1992] have shown the typical range of tx to be from 2 to 3, i.e., 2 _< D _<2.5. The phase of the Fourier components must be random. A Gaussian random number generator is usually employed to provide random numbers [e.g. Saupe, 1988; Brown, 1995]. We have found that the probability densities of surface heights for the rocks used in this study were normally distributed, although the apertures were not. This concurs with the findings of Brown [1995] for surface heights. However, Power and Tullis [1991] have concluded that there is no good surface height distribution of natural fracture surfaces because of nonstationarity of a fractal surface. While this is true in theory, we are able to define a mean for our fractal surfaces and aper-

5 GLOVER ET AL.: SYNTHETIC ROUGH FRACTURES IN ROCKS ;,.-.. ;..., = ß.,., -.' : ': ?--.. z /z: ;. ; ?.'... ;.:.-.. ' d % ' '.' ': % : :... : ,..., /..- - ::.'....- ' '...;. '..-./....' / ;. '-- :.....:. -...' , :...: ',;". :. ; -(-,., ß 4... '%'?"': :':... ' - 1.o 1.o Figure 4. The effect of changing the anisotropy factor of fractal surfaces generated on a 1024x1024 matrix. Each surface has D = 2.40 and a similar, although arbitrary, distribution of surface elevations. They differ only in the anisotropy factor: (a) highly anisotropic transversely to x (a/b=o. 1), (b) isotropic (a/b=l), and (c) highly anisotropic parallel to x (a/b=10). tures, and use a specific surface height distribution, because of surface heights in the x and the y directions [Brown, 1995]. In long-wavelength matching of the surfaces and the finite size of this modification the wavenumbers for each of the lateral dithe synthetic fractures we generate, as discussed above. A Gaussian random number generator has been used in this work. However, it is possible to use random deviates from other standard statistical distributions if, for example, a chisquare distribution of surface heights is required. If the surface is to bear comparison with a natural fracture surface, it is also necessary to scale the surface in the x and y directions. The numerically synthesized fractures created in this work were scaled in the z direction so that the standard rections, kx and k,, are treated independently, and each is scaled by a factor, a and b, respectively. The ratio a/b controls the anisotropy. If a/b < 1, anisotropy is transverse to the x direction; if a/b - 1, the surface is isotropic; and if a/b > I anisotropy is transverse to the y direction. Anisotropy is referenced to an arbitrary direction for natural rock fracture surfaces but is referenced to the x-y coordinate system for numerically synthesized fractures. Figure 4 shows the effect of changing the anisotropy factor while keeping all other surface deviation of their surface heights matched standard deviations parameters constant. of surface heights obtained from profiling natural fractures, and were scaled in the x and y directions so that the final numerical synthetic fractures were the same size as the grid of profiles taken in the profiling measurements. It is also possible to introduce modifications into the algorithm for inducing anisotropy of the standard deviation of 5. New Modification for Controlling Mismatch 5.1 Analysis of the Brown Method The method outlined above can be used to numerically synthesize fracture surfaces that have geometric parameters

6 GLOVER ET AL. ø SYNTHETIC ROUGH FRACTURES IN ROCKS '., ,. ; Y -¾ o.o o.o Figure 5. Two synthetic fracture surfaces, 1024x1024, scaled to match the physical dimensions of the original profiling data, with a lateral resolution of 0.04 mm. Eac has an isotropic fractal dimension, D = 2.4, and an isotropic distribution of surfaces heights, (5 = 1.29 mm. They are fully correlated long wavelengths (using the Brown [1995] method) resulting their initial similarity to the eye, but are independent at wavelengths shorter than 0.25 mm matching those of natural fractures found by profiling. Two such surfaces can be used to make a synthetic fracture. For such a fracture to be realistic, it must contain some degree of matching at long wavelengths [Brown et al., 1986; Brown, 1995; Matsuki et al., 1995]. Recently, Brown [1995] modigeothermal hot dry rock field in northern Japan (Table 1). In each case the top diagram (Figures 6a and 7a) shows data from the profiling measurements, and the bottom part of the diagram (Figures 6b and 7b) shows data from numerically synthesized fractures that have been created to imitate the fied the code of Saupe [ 1988] to obtain a routine that was able measured fracture. At small wavelengths, where the two surto include long-wavelength matching by controlling the random phase element in the Fourier components. In the modified faces are unmatched, the Brown code provides a good imitation of the synthetic aperture. However, it should be noted that code the phases of the Fourier components representing for the Brown code, both the PSD (Figure 6b) and the ()0 rawavelengths greater than a specified cutoff mismatch wave- tio plot (Figure 7b) contain a sharp discontinuity that is not length (say,,.(2)) were randomly chosen but identical for both present in the corresponding plot for the profile data (Figures surfaces. This resulted in the two surfaces being completely 6a and 7a). This is the result of using zi single-threshold length matched at large wavelengths. In the Brown method the phase of the Fourier components for wavelengths below the cutoff were randomly generated using completely independent generators making the two surfaces totally unmatched. This produced synthetic fractures that were much better simulations of those measured in rocks, compared to synthetic fractures created using two completely unmatched surfaces with the same geometric parameters, at least when judged by eye. However, the development of matching in these synthetics is sudden at the defined wavelength. Figure 5 shows two isotropic fractal surfaces each with a scale to model what is a gradual process in natural fractures. Furthermore, at large wavelengths each curve takes values that are much too small compared with the profile data. This arises because the Brown code assumes that, at wavelengths above the threshold length scale, the two surfaces are completely matched. In reality, this is not so because the two surfaces retain some degree of independence that decreases only gradually as wavelength increases. Unfortunately, if the Brown code is used to produce synthetic fractures, it results in underestimations of the standard deviation and mean fracture aperture and cannot be used reliably for modeling. fractal dimension of 2.40 generated using the Brown code [Brown, 1995]. At first, the surfaces look identical, but close 5.2. New Method for Controlling Fracture Mismatch inspection reveals small differences. This is due to the fractures being partially correlated at large wavelengths but being The problems with the Brown code arise from (1) using a independent at short wavelengths. sharp cutoff mismatch wavelength in' the creation of numeri- Figures 6 and 7 show the PSD and 5,(X) ratio for surfaces cally synthetic fractures when the degree of matching varies and apertures created with the Brown code and geometric parameters obtained from profiling measurements across a fracsmoothly in natural fractures, and (2) assuming that the two surfaces are perfectly matched (have identical Fourier compoture in siliceous tuff obtained from the Hachimantai model nents) at >,.. We have implemented a new way of control-

7 GLOVER ET AL.' SYNTHETIC ROUGH FRACTURES IN ROCKS 9615 loo E-6 1 E E-4 1E-6 1 E-8 (a) S. urfaces Aperture Profiling Data (b) Surfaces Aperture: With Variable Mismatching {this work) " ' Aperture: With Single Cut-off Mismatching (Brown [1995] code) Synthetic Data I spatial Frequency, 1/mm Figure 6. Power spectral densities of surfaces and apertures as a function of spatial frequency (a) from profiling data in two perpendicular directions across a fracture in Hachimantai siliceous tuff, (b) of synthetic fractures created using the geometric parameters obtained from the profiling data with the algorithm of Brown [1995] and with the method presented in this work. 2a(c2;; ß (2) Here we call [3 the "roll-off parameter". This parameter represents the maximum fractional matching achieved at the largest wavelengths and controls the extent to which the PSD of the resulting aperture flattens off at large wavelengths. In (2), k. is the wavenumber of the mismatch wavelength (.), where k.=2n/,.. The 2 in the denominator of (2) results from our arbitrary requirementhat the gradual development of matching starts at twice the wavenumber than that corresponding to k. in order that matching is sufficiently well developed to be noticeable at this value. The random number R3 is partially correlated to R1 and results in partial matching to a degree defined by [3 and k. for any given Fourier component of wavenumber k. Finally, to obtain the Fourier components of the partially matched wavelengths that are greater than kj2, we use random number R1 for the upper surface and R3 for the lower surface. An example of the use of this method is given in Figures 6b and 7b and can be compared with the results from using the Brown code in the same figures. It should be noted that the problems encountered previously by the Brown code are no longer present, and the PSD of both surfaces and apertures, as well as the (,) ratio plot reproduce those of the measured fracture (Figures 6a and 7a) very well. Additionally, the probability density plots for both the synthetic surfaces and apertures created using this modification are also Gaussian, well ling the degree of mismatch between the two numerically 10 synthesized fracture surfaces that compose the fracture. This (b) Aperture: With method avoids both problems and, in doing so, can produce Variable Mismatching fractures with PSDs and ( ) ratio plots that are statistically (this work), M v% -"m"t'" / identical to those found in real fractures, while leaving the statistical geometric properties of each surface unchanged. We use a single mismatch wavelength. to control the fre- 0.1 quency at which the matching begins to be well developed, like Brown [ 1995]. However, we do not employ this value as a 0.01 / sudden change between the use of two independent random number generators for each surface to a single random number perture: With Single / Cut-off Mismatching generator for both surfaces. In our case, we do the following: _ / (Brown [1995] code) 1. For <XJ2 we use two independent random number generators (R1 for the upper surface and R2 for the lower Synthetic Data surface). This ensures that the two fracture surfaces are inde- 1E-4... pendent (completely unmatched) in this wavelength range I For >./2 we mix R1 and R2 linearly as a function of Spatial Frequency, 1/mm wavenumber k to form a third random number, R3, according Figure 7. Ratio of the PSD of apertures to that of surfaces as a to function of spatial frequency (a) from profiling data in two per- R3 = 7 R1 + (1-7)R2, (1) pendicular directions across a fracture in Hachimantai siliceous tuff, (b) of synthetic fractures created using the geometric paramewhere 7 is a function of the wavenumber of each Fourier component, ters obtained from the profiling data with the algorithm of Brown [1995] and the with the method presented in this work. 10 o (a) Profiling Data

8 9616 GLOVER ET AL.: SYNTHETIC ROUGH FRACTURES IN ROCKS Table 1. Tuning Parameters for Synthetic Modeling Parameter Length of profile in x and y directions, mm Size of fracture array in x and y directions Lateral resolution in x and y directions, mm Fractal dimension of upper surface in x direction Fractal dimension of upper surface in y direction Fractal dimension of lower surface in x direction Fractal dimension of lower surface in y direction SD of upper surface asperity heights in x direction, mm SD of upper surface asperity heights in y direction, mm SD of lower surface asperity heights in x direction, mm SD of lower surface asperity heights in y direction, mm Mismatch length scale,,.o)?, mm Mismatch length scale,c 2)?, mm Start of mismatching, mm Roll-off parameter 13 Obtained by Profiling Used in Creation of Synthetic Fracture Not used Undefined Undefined 0.60 The results of profiling across a fracture in a sample of Hachimantai siliceous tuff is shown in the left column. The data used to tune the synthetic fracture to match it is shown to the right. J- As defined by Brown [ 1995]. correlated to those obtained from the profile data, and consistent with the standard deviations of surface heights, aperture widths, and mean apertures obtained from surface profiling of the original rock (Table 1). The fracture described by the data in Figures 6b and 7b is shown in Figure 8. Note that the two surfaces show correlation at long wavelengths but there are differences at short wavelengths, which leads to the loss of long-wavelength information in the aperture. It is difficult to compare quantitatively the results from the numerically synthesized fractures with those from the profiled data in Figures 6 and 7. Instead, we have used a cross-plot of the PSD of the numerically synthesized fracture aperture against the PSD of the aperture from profiling measurements to examine the extent to which there is a 1:1 correlation (Figure 9a). The other parts of Figure 9 show similar plots for fractures in three other rock samples for comparison. In all cases the Brown code grossly underestimates the PSD of the synthetic fracture when compared with the PSD of the aperture to which it was tuned to mimic. The discrepancy arises only at wavelengths greater than the mismatch wavelength. In the case of the Hachimantai tuff and Darley Dale sandstone, underestimations of up to 3 orders of magnitude occur, whereas for the two granite samples the underestimation is approximately 5 orders of magnitude at the longest wavelengths. By comparison, the new code mimics the PSD aperture data from profiling measurements very well over the entire wavelength range. Further information concerning the four rock samples is given in Table Determination ofkc and 15 Although it is possible to calculate k. from the value ). estimated from profiling data as did Brown [1995], we prefer to obtain the values of k. and 3 by varying these parameters until the PSD ratio and PSD curves fit their observed counterparts to the greatest extent. In practice, this is done using an iterative processes involving the following steps: (1) initial estimation of k. from k? obtained from profiling data using the Figure 8. Example of a synthetic fracture, 1024x1024, lateral resolution 0.04 mm, fully tuned to the Hachimantai fracture. The top plane is the aperture, and the lower two are the fracture surfaces. The distance between the fracture surfaces has been artificially increased for ease of viewing.

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10 ß GLOVER ET AL.: SYNTHETIC ROUGH FRACTURES IN ROCKS Table 2. Fracture Data for Various Rock Samples Rock Type Mean Grain Mismatch Mismatch Roll-off Size, mm Wavelength Wavelength Parameter, c(1), mm c(2), mm Hachimantai tuff Darley Dale sandstone Lochnagar granite Inada granite fluid flow through such a fracture precisely because the two surfaces composing the fracture are to some degree matched. (3 ) ratio plot, and estimation of an initial value for [ ;(2) calculation of a set of numerically synthesized fractures using Additionally, structure at short wavelengths is also unlikely to these values; (3) calculation of the mean stacked PSD of the have a great effect upon the mechanical and fluid flow characteristics of the fracture because the fractal nature of the suraperture of the resulting synthetic fractures; (4) correlation of the mean stacked PSD of the aperture of the resulting synthetic faces and the aperture ensure that variation of asperity height fractures with the PSD of the aperture from profiling measand aperture at short wavelengths is correspondingly very small. The range of wavelengths that has greatest potential for urements; (5) choice of new values for k,. and [, followed by a perturbing the mechanical and fluid flow characteristics of the reiteration of the process beginning at step (2). The aim of this fracture is the middle range. It is only in this range that unprocess is to maximize the correlation coefficient between the mean stacked PSD of the aperture of the resulting synthetic matched asperities of sufficient size to enable mechanical propping of the fracture and increases the tortuosity of fluid fractures and the PSD of the aperture from profiling measurements. In our case the choices of new values for k,. and [ were flow are possible. This is the scale range where fracture mismatching begins to develop. Thus the parameters [ and k,., done by hand, but clearly, this process could be improved by which describe the development of matching in this scale making the iteration process fully automatic with its terminarange, also have the potential for being used as parameters to tion dependent upon reaching some specified target correlation coefficient. However, the presence of two variables makes compare with the mechanical properties and fluid flow characteristics of fractures. automatic fitting difficult to implement in practice. We have noted that the best value for k. after the fitting procedure rarely departs significantly from the initial input, providing the estimated value of 3. (2) from the (3 ) ratio plot is based on suffi- loo cient length of profiling data. The iteration procedure is com- (a) putationally intensive and time consuming but ensures that the E 1.o-- synthetic fracture is statistically identical to the experimentally profiled data from the rock fracture. - O.Ol-,, Varying kcrnoves ß. the IocatiOe 5.4. The Roll-off Parameter - ' 164_ Note particularly that the value of the roll-off parameter [ is Varying [3 alters the ' for the Hachimantai siliceous tuff. This indicates that even the sharpness of the bend (or "knee") that both the PSD aperture curve and the (3 ) undergo without affecting the spatial frequency at which the bend (or knee) occurs. By comparison, changing the value of k. moves the spatial frequency of the knee without affecting its sharpness. It is interesting to note that the inclusion of the concept of a roll-off parameter in the creation of synthetic fractures not only enables better synthetic fractures to be created, but may also be of use as a fracture characteristic that could be correlated to the mechanical and hydraulic properties of the fracture. Longwavelength surface structure is unlikely to have any great control over the mechanical normal closure of a fracture or sharpness of the knee " rtures at the largest spatial wavelengths the two fracture surfaces are interrelated by only 60%. The roll-off parameter is some measure of the correlation between the two surfaces; however, g- I I i it is unclear exactly how it is related to the formal statistical 10 correlation between the two surfaces. Nevertheless, the roll-off (b) Varying location kcrnoves of the knee the_ I' --I parameter may be used as a new quantitative measure of the degree to which correlation of fracture surfaces develops within a rock mass. In the long-wavelength limit, as k- 0, (2) becomes ¾ = [, and (1) can be written as R3 = [ R1 + (1-[ )R2. Thus [ repß,,,/ resents the fractional control that R1 has on the final fracture aperture at the largest scale possible rather than the largest / Varying [ alters the / sharpness of the knee scale used in the application specific profiling and modeling 163- (i.e., in the ) - oo limit), whereas the parameter ¾ represents the degree of interrelation at some specific wavenumber. In the I I I case of the Hachimantai tuff [ = 0.6 represents the degree of interrelation of the two surfaces as 3 - oo. However, it is pos- Spatial Frequency, 1/mm sible that the roll-off parameter also varies with scale. 100 The effect of varying the values of the roll-off parameter Figure 10. Schematic diagrams of the effecthat varying k,. and [ and the value of k. is shown schematically in Figure 10. De- have on (a) the PSD of the synthetic fracture aperture, and (b) the creasing (or increasing) the value of [3 increases (or decreases) (3 ) ratio plot of the synthetic fracture.

11 GLOVER ET AL.: SYNTHETIC ROUGH FRACTURES IN ROCKS 9619 Table 2 indicates that, for the small set of samples presented in this work, the roll-off parameter decreases almost linearly as grain size increases. This implies that rocks with smaller grain size do not develop well-matched fractures as readily as those with larger grain sizes, providing the fractures are fresh and unaltered by subsequent mechanical or fluid abrasion. Table 2 also indicates that the mismatch wavelength increases as the grain size increases, as postulated above. For the Hachimantai tuff and the Darley Dale sandstone it is that provides the best estimate of the grain size, but for the Lochnagar and Inada granites, ).( ) gives the better estimate of grain size Application to Rock Property Modeling If all the geometrical parameters of the natural fracture listed in Table 1 are found, we can use the model presented in this paper to generate any number of different synthetic fractures which mimic well the original rock fracture by using different initial random generator seeds. A population of such fractures can then be used in mechanical, hydraulic, electrical, and heat flow modeling with confidence that they share the same gross geometrical properties of the natural fracture to which they are tuned, but offering two advantages. The first advantage is that a large population of physically distinct, but statistically identical, fracture models can be examined. The second advantage is that the effect of systematically varying the geometrical parameters that are used to create the fractures on the examined rock property can be carded out. We have carried out fluid flow modeling using a population of synthetic fractures created with the code described in this paper. This work is reported in a companion paper [Glover et al., this issue]. 6. Summary A new spectral synthesis controlled mismatching method for creating synthetic fractures with rough surfaces has been developed. The new approach, which incorporates a gradual onset of mismatch between surfaces rather than the sudden onset used in earlier techniques [Brown, 1995], provides a much closer match between observed and synthetic fractures. The new approach requires the following parameters to be obtained in order that the resulting synthetic fracture closely imitates that in the rock. 1. Fractal dimensions of each surface, obtained from profiling data. 2. A measure of the mismatch wavelength (e.g.,)?), obtained from profiling data. 3. The roll-off parameter obtained by reiterative fitting of the resulting synthetic fast Fourier transform (FFF) data to FFr data from profiling. 4. Standard deviations of the asperity heights on each surface in each lateral perpendicular direction, obtained from profiling data.. 5. Lateral dimensions of the fracture (i.e., lateral dimensions used in the profiling measurements). Parameters 1 and 2 can be obtained easily from the Fourier analysis of profiles taken across rock fractures. Parameter 3 is obtained by fitting the power spectral density of the synthetic aperture to that of the profiled data. Parameter 4 is procurable easily from the statistical analysis of the same data set, and parameter 5 is implicit in the profile data. The availability of a code that enables high-quality synthetic fractures to be created has great application in the modeling of the mechanical and transport properties of rocks containing fractures. The systematic variation of the parameters defining the synthetic fractures can be used to explore the effect each has upon the physical properties of the rocks containing such fractures. The use of synthetics enables this work to be done without recourse to extensive time-consuming profiling. Additionally, a population of synthetic fractures that each have identical statistical geometric properties but are physically distinct can be created by altering only the two random number seeds used to generate each fracture. Modeling on such a population of fractures provides results where random errors in the modeling associated with unrepresentative fractures are reduced in an analogous manner to carrying out profiling measurements on a suite of natural fractures rather than relying upon just one. Aclmowledgments. The authors would like to thank William Durham and William Power for their helpful review suggestions, as well as Takatoshi Ito, Jonathan Willis-Richards, and Rod Stewart for their helpful comments. Many of the models reported in this work were developed on the CRAY-916 of the Institute of Fluid Science Supercomputer Centre, and the NEC SX-3 of Tohoku University Supercomputer Centre, both in Sendai, Japan. References Aharonov, E., and D.H. Rothman, Growth of correlated pore-scale structures in sedimentary rocks: A dynamical model, J. Geophys. Res., 101, , Amadei, B., and T. Illangsekare, A mathematical model for flow and transport in non-homogeneous rock fractures, Int. J. Rock Mech. Mits. 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12 9620 GLOVER ET AL.: SYNTHETIC ROUGH FRACTURES IN ROCKS Mandelbrot, B.B., The Fractal Geometry of Nature, 486 pp., W.H. Free- Thompson, A., Fractals in rock physics, Annu. Rev. Earth Planet. Sci., 19, man, New York, , Matsuki, K., T. Kojima, and T. Murai, Surface roughness and initial aper- Thompson, A., A. Katz, and C. Krohn, The microgeometry and transport ture distribution of a small-scale hydraulic fracture in granite (in Japa- properties of sedimentary rocks, Adv. Phys., 36, , nese with abstract and diagrams in English), J. Geotherm. Res. Soc. Wang, J.S.Y., and T.N. Narasimhan, Hydrologic mechanisms governing Jpn., 17, , fluid flow in a partially saturated, fractured, porous medium, Water Re- Matsuki, K., J.-J. Lee, and T. Kojima, A simulation of the closure of a sour. Res., 21, , small-scale hydraulic fracture in granite (in Japanese with abstract and Wang, J.S.Y., T.N. Narasimhan, and C.H. Scholz, Aperture correlation of a diagrams in English), J. Geotherm. Res. Soc. Jpn., 18, 27-37, fractal fracture, J. Geophys. Res., 93, , Peitgen, H-O., and D. Saupe (Eds.), The Science of Fractal Images, 445 Watanabe, K., and H. Takahashi, Fractal geometry characterization of pp., Springer-Verlag, New York, geothermal reservoir networks, J. Geophys. Res., 100, , Power, W.L., and T.E. Tullis, Euclidean and fractal models for the descrip- Yoshioka, N., and C.H. Scholz, Elastic properties of contacting surfaces tion of rock surface roughness, J. Geophys. Res., 96, , under normal and shear loads, 1; Theory, J. Geophys. Res., 94, 17,681- Power, W.L., and T.E. Tullis, The contact between opposing fault surfaces 17,690, 1989a. at Dixie Valley, Nevada, and implications for fault mechanics, J. Geo- Yoshioka, N., and C.H. Scholz, Elastic properties of contacting surfaces phys. Res., 97, 14,425-14,435, under normal and shear loads, 2; Comparison of theory with experi- Power, W.L., T.E. Tullis, S.R. Brown, G.N. Boitnott, and C.H. Scholz, ment, J. Geophys. Res., 94, 17,691-17,700, 1989b. Roughness of natural fault surfaces, Geophys. Res. Lett., 14, 29-32, Zimmerman, R.W., S. Kumar, and G.S. Bodvarsson, Lubrication theory analysis of the permeability of rough-walled fractures, Int. J. Rock Pyrak-Nolte, L.J., N.G.W. Cook, and D.D. Nolte, Fluid percolation through Mech. Min. Sci. Geomech. Abstr., 28, ,1991. single fractures, Geophys. Res. Lett., 15, , Ripley, B.D., Spatial Statistics, p. 10, John Wiley, New York, Russ, J.C., Fractal Su.rfaces, 309 pp., Plenum, New York, P. W. J. Glover, Department of Geology and Petroleum Geology, Sammonds, P.R., P.G. Meredith, S.A.F. Murrell, and I.G. Main, Modelling University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotthe damagevolution in rock containing a pore fluid by acoustic emis- land. ( p.glover@abdn.ac.uk) sion, in Rock Mechanics in Petroleum Engineering, Proceedings of K. Hayashi, Institute of Fluid Science, Tohoku University, EUROCK '94, SPE/IS International Conference, De.lft, The Nether- Katahira, Aoba-ku, Sendai , Japan. ( hayashik@ lands,pp , A.A. Balkema, Brookfield, Vt., ifs.tohoku.ac.jp) Saupe, D., Algorithms for random fractals, in The Science of Fractal Im- R. Hikima and K. Matsuki, Department of Geosciences and Techages, edited by H.-O. Peitgen and D. Saupe, pp , Springer- nology, Tohoku University, Sendai , Japan. ( Verlag, New York, hiki@rock.earth.tohoku.ac.jp; Schmittbuhl, J., F. Schmitt, and C.H. Scholz, Scaling invariance of crack surfaces, J. Geophys. Res., 100, , Swan, G., and S. Zongqi, Prediction of shear behaviour of joints using pro- (Received December 6, 1996; revised July 12, 1997; files, Rock Mech. Rock Eng., 18, , accepted October 1, 1997.)