{'fraction_non_alphanumeric': 0.03692324277705908, 'fraction_numerical': 0.01476929711082363, 'mean_word_length': 4.5731450886152, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A geometrical aperture-width relationship for rock fractures

The relationship between fracture aperture (maximum opening; dmax) and fracture width (w) has been the subject of debate over the past several decades. An empirical power law has been commonly applied to relate these two parameters. Its exponent (n) is generally determined by fitting the power-law function to experimental observations measured at various scales. Invoking concepts from fractal geometry we theoretically show, as a first- order approximation, that the fracture aperture should be a linear function of its width, meaning that n = 1. This finding is in agreement with the result of linear elastic fracture mechanics (LEFM) theory. We compare the model predictions with experimental observations available in the literature. This comparison generally supports a linear relationship between fracture aperture and fracture width, although there exists considerable scatter in the data. We also discuss the limitations of the proposed model, and its potential application to the prediction of flow and transport in fractures. Based on more than 170 experimental observations from the literature, we show that such a linear relationship, in combination with the cubic law, is able to scale flow rate with fracture aperture over ~14 orders of magnitude for variations in flow rate and ~5 orders of magnitude for variations in fracture width.


Introduction
Modeling flow and transport within fracture networks requires knowledge of fracture attributes and their scaling properties. A fracture is typically characterized by its aperture, width, displacement, and surface roughness. It is well documented in the literature that the surfaces of natural rock fractures are rough and follow self-affine scaling from fractal geometry [1][2][3][4] . The self-affinity of the surfaces of fractures is best understood in a statistical sense 5 . According to self-affinity, if one considers the height difference between two points separated by distance |xi -xj| on a self-affine surface, then 〈ℎ( % ) − ℎ( ( )〉 ∝ + % − ( + , (1) where h(xi) and h(xj) are the heights at points xi and xj, respectively, and H is the Hurst exponent characterizing the surface roughness. Generally speaking, the larger the Hurst exponent, the smoother the fracture surface. A value of H ~ 0.8 was reported for granite 6 , while H ~ 0.5 for sandstones 7 . However, a wider range of the Hurst exponent e.g., 0 < H < 0.9 has been reported for a variety of rock joints 5 .

3
One of the most important characteristics of an individual fracture is its aperture, defined herein as the maximum opening (dmax). The maximum opening is related to the average opening via ./0 = 4⁄ /56 8 . Although aperture has been typically characterized as either the maximum or the average opening, due to the rough surface of fractures, there exists a distribution of aperture openings, rather than a single unique value for a given fracture. It is worth mentioning that the aperture opening distribution can only be inferred once the fracture surfaces have been characterized, which requires accurate imaging of the surface roughness.
Fracture aperture has been shown to control fluid flow and solute transport processes. For example, the well-known cubic law 9,10 relates flow in a single fracture to the product of fracture width w and fracture aperture raised to the power three (i.e., ∝ /56 9 ∇ℎ ∝ ./0 9 ∇ℎ in which Q is flow rate and ∇ℎ is hydraulic gradient).
Accordingly, the scaling relationship between fracture aperture and its width has been the subject of active research and debate over the past several decades. We should point out that the terms width and length have been interchangeably used in the literature to describe the straight line from one tip of a rock fracture to another, which can be a source of confusion.
A commonly applied model linking fracture aperture to fracture width is the following empirical power law 11,12 : ./0 = < (2) in which c is a constant coefficient, dmax is the fracture aperture as previously defined, w is the fracture width, and n is an empirical exponent. Different values of n have been reported in the literature based on experimental observations, stochastic reasoning, and theoretical models. In Table 1 Those authors also showed that n = 1 could accurately represent their measured data.
Besides power law scaling 16,17 , the log-normal probability density function has also been used to describe distributions of fracture aperture and/or width 18,19 .  Other values of n, estimated by fitting data sets obtained over a variety of length scales, are listed in Table 1. Interestingly, the arithmetic average of all of the experimentally-determined n values in Table 1 is 1.05±0.07, a value which is not 6 statistically different from unity. It should be noted that most n values presented in Table   1 are based on rock fracture measurements. However, n = 0.47 ± 0.03 was derived by Walmann et al. 23 from cracks in clayey soils, which are unconsolidated as compared to rocks.
The main objective of this study is to use concepts from fractal geometry to develop a first-order approximation of the relationship between fracture aperture dmax and fracture width w. To our knowledge, no such approach has previously been proposed to predict the dmax-w relationship. In what follows, we briefly describe fractal geometry, introduced by Benoit B. Mandelbrot 24 , and present its fundamental concepts. We then derive a geometrical relationship between the aperture and width of a single rough-walled fracture, resulting in a geometrically-based prediction of the exponent n in Eq. (2).
Finally, we compare our theoretical results with experimental observations reported in the literature and relate them to flow and transport models.

Theory
Fractal geometry, introduced by Benoit B. Mandelbrot 24 , has been shown to be a robust and appropriate approach for modeling the multiscale structure of complex and heterogeneous media such as rocks, soils, and fracture networks 5,25-32 . A fractal object is characterized by having a (typically non-integer) dimension less than the Euclidean dimension of the space it is embedded in. If a fractal object is rescaled in all directions with the same scaling factor, a statistically similar object is reproduced; this property is termed self-similarity.
Many natural objects are self-affine rather than self-similar, meaning that they require different scaling factors in different directions 5 . The concept of self-affinity has been used to model the physical and geometrical properties of fracture networks 4,14,33 . It has also been applied to statistically characterize fracture surfaces 34 . Poon et al. 35 , among others (see Sahimi 5 , for a comprehensive review), modeled surface roughness of fractures by means of self-affinity. However, none of these models predict the relationship between fracture aperture dmax and fracture width w.
In the simple model that we present, self-affinity is assumed. Let us presume that the cross-section of a natural fracture can be represented by an ellipse with a rough boundary, as shown in Fig. 1. It should be pointed out that an unembroidered ellipse is the shape of a Griffith crack traditionally employed in linear elastic fracture mechanics (see e.g., Zimmerman and Main 36 ). Other crack geometries e.g., edge, corner, or semicircular have also been used within Griffith theory. We further assume that the rough boundary is fractal, and thus, following Mandelbrot 25 , one may relate the fracture perimeter P to its area A as follows: where Db is the boundary fractal dimension (1 ≤ Db < 2) characterizing the roughness of a cross-section taken through a fractal fracture surface (Fig. 1). The higher the Db value, the rougher the boundary.
Equation (3) has been successfully applied to relate perimeter to area in clouds 37 , metallic rough fractures 38 , and soil and rock pores 39,40 . Furthermore, it has been also used in combination with the model of Patzek and Silin 41 to describe fluid flow in tubular pores with rough surfaces, and to accurately estimate water relative permeability 42 . 8 We can compute the width of the fractal perimeter of a rough-boundary ellipse shown in Fig. 1, by invoking the Mandelbrot 25 approach, followed by Wheatcraft and Tyler 43 and many others. Based on this approach, a fractal length Lf is a function of some measurement scale and the straight-line distance Ls between the two ends of the fractal path as follows: in which Dl is the fractal dimension of the fractal length. Equation (4) is valid for both self-similar and self-affine fractal curves. For a self-similar fractal curve, divider method, box counting, and mass scaling estimates of Dl will all be the same. However, for a selfaffine fractal curve, different values of Dl will be obtained using different evaluation methods 44 . We should also note that the concept underlying fractals is self-similarity or self-affinity, that is, invariance against variations in scale or size (scale-invariance).
Accordingly, the fractal dimension is theoretically scale-invariant. However, in the nature, objects are only approximately fractal and scale invariant, and one may expect the fractal dimension to vary from one scale to another. A notable example is fractal dimension determination from images. For example, Baveye et al. 45 demonstrated the effects of image resolution, thresholding, and algorithm used to generate binary images on the estimation of fractal dimension.
Assuming that the rough boundary shown in Fig. 1 is a fractal length, setting Dl = Db and Ls = w in Eq. (4) in combination with ∝ U yields: Equation (5) necessarily means that perimeter is measurement scale-dependent. Given that 1 ≤ Db < 2, P is directly proportional to the fracture width w, while inversely proportional to the measurement scale . As a consequence, when tends to zero, P approaches infinity.
The area of an ellipse with a smooth boundary (represented by the black dashed line in Fig. 1) is a function of the product of its semi-minor and semi-major axes ( ./0 ). Accordingly, for the rough-boundary ellipse (shown in red) one can approximately set Note that Walmann et al. 23 also used a relationship similar to Eq. (6) to relate the area of a fracture to its aperture and width.
Substituting Eqs. (5) and (6) into Eq. (3) gives where C is a numerical prefactor whose value depends on the measurement scale (i.e., Equation (7), represents a geometrical scaling relationship linearly relating fracture aperture to fracture width. It is in agreement with the linear elastic fracture mechanics (LEFM) approach, which predicts that the fracture aperture should scale linearly with fracture width 13 . Within the LEFM theory framework, the coefficient C would be equal to (1 − )/ in which is the effective driving stress (remote tension plus the internal fluid pressure), is Poisson's ratio, and is the shear modulus 13 .
As the derivation of Eq. (7) shows the roughness exponent cancels meaning that it is mainly the fracture geometry governing the linear dmax-w relationship rather than the roughness per se. It is interesting that the rough-boundary ellipse produces the same result as the smooth-boundary ellipse applied by LEFM. Although our geometrical terminology is different from that of LEFM, the obtained results indicate that perturbing the boundary of a smooth-boundary ellipse, used in LEFM theory, with self-affinity duplicates the linear w-dmax relationship. In Appendix A, using the same geometrical terminology, we, however, demonstrate that there exists no simple linear relationship between fracture aperture and its width for a smooth-boundary ellipse, unless w is significantly greater than dmax. In Appendix B, we assume a rectangular fracture embroidered with a fractal rough boundary (see Fig. B1) and demonstrate that when ≫ /56 the average fracture aperture (dave) should scale linearly with its width (w).
Equation (7) is also in agreement with the stochastic approach of Scholz and Cowie 15 and the average of the experimentally-determined n values presented in Table 1.
However, there are some exponents reported in the literature (e.g., those from Hatton et al. 20 ) that apparently do not match the theoretical prediction of n = 1. In what follows, we compare the geometrical model with two datasets from Hatton et al. 20 and demonstrate that Eq. (7), with different numerical prefactor C values, is able to accurately match the experimental measurements by Hatton et al. 20 .

Comparison with Hatton et al. experiments
Hatton et al. 20   Kelduhverfi is less than the 79 stated in Hatton et al. 20 . The difference between the two might be due to low figure quality or duplications in the measurements.
Fracture aperture as a function of fracture width is shown on a log-log scale for the two areas in Fig. 2. As can be observed in Fig. 2a, the data points are highly scattered, not only at short length scales but also over the entire range of length scales. For example, at short length scales a fracture aperture of 0.001m corresponds to a wide range of fracture widths from near 0.1 to 1m, almost one order of magnitude. Such scatter in the data causes substantial uncertainties in the optimization of any mathematical function's parameters through a direct fitting process.
In the Kelduhverfi dataset (Fig. 2a)  Our results presented in Fig. 2 Fig. 1), similar to Eq. (6). In their Fig. 2, however, they discuss that for some experiments the value n = 0.5 might be more relevant than n = 1.
Interestingly, Schultz et al. 46 found that the same numerical prefactors C = 0.001, 0.01, and 0.1 could accurately capture aperture-width trends for more than 170 fractures from 14 datasets including normal faults, strike-slip faults, and thrust faults.

Comparison with flow rate experiments
The correlation between aperture and width has an impact on flow rate Q in a single fracture, which can be approximated by the following cubic law 9,10 and the values n = 0.5 8,46,48 and n = 1 (see Table 1   in which DT represents the tortuosity fractal dimension. Given that theoretically 1 ≤ j < 3, the exponent 6 -DT can be expected to range between 3 and 5, with j = 2 corresponding to our model prediction. Further research on comparing these different approaches could be valuable.

Discussion
One of the main assumptions in our theoretical derivation is that the boundary of the fracture surface is fractal. Many studies in the literature indicate that fracture surfaces are rough and obey fractal geometry 1,2,52-55 . However, not every rock fracture surface is necessarily fractal at all scales. In fact, natural fractures that exhibit self-similarity or selfaffinity might lose their fractal properties above and below upper and lower cutoff scales.
The aperture-width relationship for fractures whose surface is non-fractal may well deviate from the theoretical linear function given in Eq. (7), as we show in Appendix A.
In addition, we assumed that fractures are elliptical with a rough boundary.
Furthermore, rather than a unique aperture for a given fracture, there exists a distribution of gap sizes due to the rough surfaces of fractures. Thus, we should caution that any significant deviation from elliptical shape may cause uncertainties in our proposed scaling relationship.
Madadi et al. 60 studied fluid flow in two-dimensional fractures with rough selfaffine surfaces, quantified with the Hurst exponent H, using the lattice-Boltzmann method. They used five values of H ranging from less than 0.5 (which generated rough surfaces) to near 1 (which provided relatively smooth surfaces). In their study, the mean aperture of the fractures was varied to generate narrow to wide fractures. They stated that, "Using a simple mean aperture for representing a fracture with rough internal surfaces will always result in gross errors, unless, of course, the fractures are wide, in which case the surface roughness does not really matter." A rough profile (i.e., the fractal path shown in Fig. 1) can be characterized by two parameters: (1) the boundary fractal dimension (or Hurst exponent), and (2) the rootmean-square of the roughness height. The former characterizes the boundary roughness, while the latter controls the roughness thickness. The higher the boundary fractal dimension, the rougher the fractal path. Likewise, the higher the root-mean-square of the roughness height, the thicker the fractal profile. In our theoretical framework, described in section 2, only the boundary fractal dimension was employed. The effect of the rootmean-square of the roughness height was not incorporated. Further investigation is required to understand how the root-mean-square of the roughness height might impact the predictions of our theoretical linear relationship, Eq. (7).

Conclusions
In this study, we developed a first-order linear approximation of the relationship relating fracture aperture (dmax) to fracture width (w

Appendix A
In this section, we consider an elliptical fracture with a smooth boundary presented in Fig. A1. We show that there exists no simple relationship between fracture aperture and its width, unless the latter is significantly greater than the former. i.e., ∝ . Given that ∝ ./0 , substituting P and A into Eq. (A1) yields which is the same as our Eq. (7). The difference between Eq. (A5) and Eq. (7) is that the former holds if and only if ≫ ./0 , while the latter does not depend on any assumption between and ./0 . Fig. A1. An elliptical fracture with a smooth boundary. dmax and w represent the fracture aperture (maximum opening) and the fracture width, respectively.

Appendix B
Here we assume a rectangular fracture with a fractal rough boundary (see Fig. B1 which is similar to Eq. (7). One should note that any fracture whose geometry differs significantly from that given in Fig. B1 and/or any failure in the assumptions ∝ V P (Eq. B2) and ∝ /56 (Eq. B3) could cause significant deviations from Eq. (B4).  Fig. 1. Schematic cross-section through an elliptical fracture with a rough boundary. dmax and w represent the fracture aperture (maximum opening) and the fracture width, respectively.    46 14 datasets from various studies (details given in their Fig. 1) -1 NA Lai et al. 65 Open fractures in tight gas sandstones 110 1 0.99

Figures caption
Calcite-filled fractures in tight gas sandstones 300 1 0.95 * n represents the exponent in the power-law aperture-width relationship (Eq. 2); NA means not available.