Mapping fracture aperture as a function of normal stress using a combination of casting, image...

$: Mapping fracture aperture as a function of normal stress using a combination of casting, image analysis and modeling techniques$
To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:3-4, paper No. 132. Published by Elsevier Science Ltd 1997

Copyright © 1997 Elsevier Science Ltd

Int. J. Rock Mech. & Min. Sci. Vol. 34, No. 3-4, 1997 ISSN 0148-9062

To cite this paper: Int. J. RockMech. &Min. Sci. 34:3-4, Paper No. 132

MAPPING FRACTURE APERTURE AS A FUNCTION OF NORMAL STRESS USING A COMBINATION OF CASTING,

IMAGE ANALYSIS AND MODELING TECHNIQUES

S y l v i e S. G e n t i e r l ; D e b o r a h L. H o p k i n s 2

1 Bureau de Recherches G~ologiques et Mini~res, DR/HGT, BP 6009-45060, Orleans, Cedex 2, France

z Lawrence Berkeley National Laboratory, MS: 46A-1123, Berkeley, CA, 94720 USA

A B S T R A C T

A combination of casting, image analysis and modeling techniques have been used to produce aperture maps of natural fractures under a normal load. Images of the fracture casts are corrected for trends introduced by the inhomogeneity of the light source, and a precision-engineered mold is used to make a calibration wedge for each fracture that is used to determine the relationship between grey-level and aperture. The resulting aperture maps are input to an analytical model that analyzes how the void spaces deform under stress. For natural fractures in granite core samples, results from the model are compared to LVDT measurements of displacement across the fracture, and maps of contact area generated via image analysis of the fracture casts. The laboratory measurements and modeling results both indicate the importance of accounting for the deformation of fracture surfaces in calculating changes in fracture aperture with stress; the aperture maps obtained via image analysis alone, assuming that aperture changes uniformly across the fracture, over-estimate contact area and do not accurately portray changes in void geometry that occur under load. The research demonstrates the important role that image processing can play in understanding and modeling the hydromechanical behavior of natural fractures, and the importance of building a methodology for transforming images into physically meaningful data.

Copyright © 1997 Elsevier Science Ltd

K E Y W O R D S

I m a g e analys i s • resin casts • f racture m o d e l i n g • fracture aper ture • h y d r o m e c h a n i c a l b e h a v i o r • f racture d e f o r m a t i o n • f luid f low • n o r m a l jo in t st i f fness • f racture c losure • pore space • h y d r a u l i c c o n d u c t i v i t y • m e c h a n i c a l proper t i e s

I N T R O D U C T I O N

Fractures can be envisioned as two rough surfaces in partial contact. The surface roughness, and typically sparse contact area, result in a void space between the fracture surfaces with a complex geometry. The mechanical response of fractures is strongly related to the amount and geometry of the contact area, while the hydrological response is governed by the connectivity and size of the voids. Changes in the state of stress in a rock mass, both natural changes, and those induced by engineering activity, are reflected by changes in the void geometry as the fractures open or close, and these changes can have a substantial affect on the flow of fluids. To predict the mechanical response of a fracture, the volume and speed of flow through the fracture, and preferential flow paths, it is necessary to understand the

ISSN 0148-9062

To cite this paper: Int. J. Rock Mech. & Mm. Sci. 34:3-4, paper No. 132. Published by Elsevier Science Ltd 1997

relationship between void geometry and hydromechanical behavior, and how it is affected by changes in s t r e s s .

Research designed to study the hydromechanical behavior of natural fractures is greatly hampered by the inability to see and characterize the morphology of the fractures. Fracture traces only hint at topography, aperture, and the roughness of the fracture surfaces which can all vary substantially over the lateral extent of the joint. For through-going fractures in laboratory samples, the specimen can be separated to expose the fracture surfaces. Various techniques have been used to measure the surface roughness and topography including laser profilometry and image analysis. However, experience has shown that it is difficult to construct the void space geometry from profilometer measurements or images because of the difficulty of determining the correct alignment when combining the data from the two separate surfaces. Laboratory methods have also been developed to determine contact area with stress (e.g. see Pyrak-Nolte et al. 1987), but again, these methods do not provide the three-dimensional geometry of the void space that is necessary for hydromechanical characterization of the fracture.

The casting methodology presented here has the advantage of duplicating the void space directly. To date, resin casts of several natural fractures have been made and analyzed via image processing to produce aperture maps under no applied load. The next step is to determine how the void spaces deform under stress. In previous work, techniques from mathematical morphology were applied to aperture maps to create skeletonized images which show the connectivity of voids at different stress levels (Billaux, Gentier 1990). These images are used in fluid flow analyses. In the current work, the aperture map obtained under zero applied load is used as the input to an analytical model that analyzes fracture deformation, contact area, and changes in aperture as a function of normal stress. For two natural fractures under a normal load, the contact area calculated with the model is compared to that obtained using an image processing technique. In conjunction with the image analysis and modeling work, laboratory experiments have also been performed. These experiments provide independent data that is used to verify results obtained from image processing and modeling.

CASTING M E T H O D O L O G Y

Over the past decade, a methodology for making casts of natural fractures has been developed and perfected (Gentier et al. 1989). Several different methods of making the casts have been explored including vacuum cycling and injection. For the work described here, a silicon polymer resin was used to create casts of natural fractures in granite core samples. A dye is mixed with the resin so that it will absorb light. After the resin is poured onto one side of the fracture, the other side is fit on top, and the resin is allowed to cure with no additional load other than that due to the weight of the sample. Once the resin has cured, the cast is removed. Particularly for tight fractures, the resin cast is very thin, and thus easily deformed and torn. To avoid these problems, a thicker layer of resin (about 1 cm) is cast to one side of the void cast.

A new method is under development that will allow the casts to be made while the sample is subjected to a normal load. The new technique has the advantage of lessening the chances for misalignment of the test specimen. Once perfected, this procedure will provide fracture casts that can be analyzed via image processing to produce aperture maps for fractures under a normal stress. At present, such maps can only be generated using the deformation model that takes as input the aperture map produced from the fracture cast obtained under zero applied load. The new method will result in data that will be used in fluid flow analyses, and to further test and validate the analytical model for fracture deformation.

ISSN 0148-9062

To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:3-4, paper No. 132. Publ i shed by Elsevier Science Ltd 1997

I M A G E ANALYSIS

Once the fracture cast has cured, a pixel map of the void space is recorded using a video camera placed above the cast while it is illuminated from below by a halogen lamp. The resulting grey-level image consists of 512 by 512 pixels, where each pixel represents an area of 0.448 mm by 0.300 mm. Translating the grey-level image into a map of void thicknesses requires two steps: removing trends in the image that are due to the inhomogeneity of the light source and establishing the correspondence between grey-level and fracture aperture.

The heterogeneity of the light source biases the digitized image. Figure 1 a is a three-dimensional plot that shows the variation in intensity over the light source. Note that intensity is highest in the center and lowest at the edges. This is evident as a halo on the images that is easily seen in Figure lb which is an image of the light source itself. For a given pixel, the grey-level recorded by the camera varies exponentially with the light power received. To obtain the corrected image, i.e. the one that would be obtained if the light source were perfectly uniform, the image of the light powers emitted (logarithm of the image of the light source) is subtracted from the image of the light powers received (logarithm of the image of the cast). This correction assumes that the images of the light source and cast are digitized using the same grid and camera settings.

A grey-level image of the cast of a natural fracture is shown in Figure 2a before correction for trends introduced by the light source, and in Figure 2b after the image has been corrected as described above. The dark grey areas correspond to large fracture apertures, whereas lighter grey areas correspond to thinner apertures. The dark circles in the center of the images are areas that were over-cored around holes that were used to inject fluids for permeability measurements made on the same samples.

To determine the relationship between grey-level and aperture, a calibration wedge is cast for each fracture using a precision-engineered mold and the same resin used for the casts of the fractures. An image of the wedge is obtained in the same way as for the fracture casts: a grey-level image is produced using a video camera and corrected for variations due to the light source (see Figure 3). The corrected image of the calibration wedge provides the necessary correspondence between grey-level and cast thickness. The resulting grey-level versus wedge thickness data are fit with a sixth degree polynomial as shown in the following graph. This function is then used to transform the grey-level images of the

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The resulting aperture maps are for the fracture under zero applied normal load; recall that the casts are made with the fracture subjected only to the weight of the sample itself. The next problem is to determine how the void space deforms under a normal stress.

CHANGES IN APERTURE UNDER A NORMAL STRESS

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Fracture aperture is defined as the thickness of the void space perpendicular to the plane of the fracture, and is zero at points of contact. As described above, image analysis provides a map of fracture apertures under zero applied load as measured from the cast of the fracture. Images of two different natural fractures in granite are shown in Figure 4. For these two fractures, two methods were employed to determine contact area and aperture maps at specified levels of applied normal stress: one based on image analysis, and one based on modeling the mechanical deformation of the fracture.

Apertures Determined via Image Analysis

In the image analysis approach, it is assumed that when a normal load is applied, the fracture closes uniformly over the area of the fracture; i.e. that the change in aperture at any point is equal to the average closure across the fracture at that stress. The fracture is represented as an array of rectangular elements where each element corresponds to one pixel. At any increment of closure corresponding to an applied normal stress, analysis of the fracture image provides an estimate of the contact area; i.e. for each element, if the amount of average closure is greater than the aperture, then the fracture is assumed to be in contact over the area of that element. At all elements not in contact, the new aperture is assumed to be equal to the difference between the initial aperture and the amount of average closure.

A problem with this method is that it does not account for changes in void geometry with increasing stress. When a fracture is subjected to a normal stress, the void area is deformed. Because the fracture surfaces are rough, this deformation is not uniform across the fracture; rather, contacting asperities tend to prop the fracture open while more than average closure occurs in open areas. Deformation of the fracture surfaces also results in less contact area than is predicted with this method. An important goal of this research is to determine the magnitude of changes in void geometry and their role in hydromechanical behavior. As described below, an analytical model describing this mechanical deformation has been developed to study changes in fracture aperture with applied normal stress.

Apertures Determined Using an Analytical Model for Fracture Deformation

The analytical model used here to analyze the mechanical deformation of the fractures is based on results from the theory of elasticity. The model accounts for deformation of the fracture surfaces, asperity deformation, and mechanical interaction between contacting asperities. Normal displacement across the fracture is calculated from the compression of the asperities, and the deformation of the fracture surfaces which accounts for most of the displacement. The theorem of least work is applied to solve for the distribution of forces that minimizes the strain energy calculated from the work done in deforming the joint. (For details and comparison of model results to analytical solutions see Hopkins 1991).

The fracture is modeled as two parallel half-spaces separated by variable height asperities. A composite topography is created by discretizing the joint and measuring the aperture at each element. The procedure is illustrated in Figure 5 for a cross-section perpendicular to the plane of the joint. Two parallel reference planes are defined by the maximum aperture as indicated in 5a. The composite surface height at each point is the sum of the upper and lower asperity heights between the reference planes, while the aperture is the distance between the reference planes minus the sum of the heights. The composite joint surface for the cross-section is shown in Figure 5b. The composite joint surface is then discretized as shown in Figure 5c. Since linear elasticity is assumed, the normal deformation of the composite surface is mechanically equivalent to the deformation of the two original joint surfaces. Since the model accounts for mechanical interaction, asperities can be placed contiguously to model rough fracture surfaces such as those shown in the images in Figure 4.

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An important feature of the model is that it accounts for deformation of the fracture surfaces as shown schematically in Figure 6. In the image analysis method described above for determining aperture changes, and in models that assume that all normal displacement is due to asperity deformation, there is no change in void space geometry under increasing normal stress (see Figure 6b). When deformation of the fracture surfaces is accounted for, there are substantial changes in void geometry as illustrated in Figure 6c.

The input to the model is material properties and a map of the fracture aperture under zero load. The aperture map is obtained from the image of the fracture cast as described above. For the results presented here, the apertures input to the model were the average values calculated over four pixels. For a specified average displacement across the fracture, the output from the model is the new aperture at each grid element, and the force acting on the elements in contact.

RESULTS

As part of a program to study the hydromechanical behavior of fractured rock, a series of laboratory experiments have been performed on core samples containing natural fractures. For each sample, four LVDTs that spanned the fracture were used to measure normal displacement for several cycles of loading and unloading. The results are shown in Figures 7 and 8 for the fractures pictured in Figure 4. Perhaps the most striking result is the difference in displacement measured by the four LVDTs on the same samples. These measurements clearly show that normal displacement varies substantially over the lateral extent of the fracture providing strong evidence of the importance of accounting for the deformation of the fracture surfaces in modeling aperture changes with stress. In addition, the results show the importance of using several LVDTs to measure displacement across the fracture to get a meaningful value of the average displacement and normal joint stiffness.

Also plotted on the graphs are the values of normal stress versus displacement calculated with the analytical model. The model results represent the average value of displacement over the entire fracture, so they should lie somewhere between the laboratory data depending on the degree to which the LVDT measurements are representative of the range of displacements across the fracture. The shape of the curves based on the modeling results differs from most of the curves of measured data in that they show more displacement at higher stresses. In other words, the model results give a lower normal joint stiffness than indicated by the LVDT measurements. The shape of these curves is determined by the rate at which asperities come into contact with increasing normal stress. It is, therefore, important that the elements used in the model be sufficiently small to adequately describe the surface topography. In future studies, the modeling work will be repeated with elements of different sizes and shapes to determine the level of discretization necessary to better match the measured data.

In Figures 9 and 10, contact area calculated from image analysis alone is compared to that calculated with the analytical model for the same average closure across the fracture. For the images based on the modeling results, the grey-levels are proportional to the normal force acting on those elements. Note that substantially less contact area is calculated using the analytical model. The fundamental difference between the two approaches that leads to the difference is that the model accounts for deformation of the fracture surfaces. An additional factor in the results presented here is that the element used in the model to represent each pixel had roughly half the area of the pixel. As discussed above, an important remaining research issue is determining what level of discretization of the fracture images is necessary to accurately predict hydromechanical behavior, and what size element should be used to represent any given area of the fracture surface in the deformation model. This is a more difficult task than it might appear because

ISSN 0148-9062


the fracture surfaces are rough at all scales, meaning that the true contact area is small and dispersed.

C O N C L U S I O N S

The research summarized here demonstrates the important role that image processing can play in understanding and modeling the hydromechanical behavior of natural fractures subjected to a normal stress. The work also demonstrates the importance of research and analysis that builds a methodology for transforming images into physically meaningful data that can be used directly, or as input to physical models. Such data reduction techniques are essential in moving forward to data analysis and modeling on a field scale. Equally important is development of independent checks on laboratory and modeling results.

The laboratory measurements and modeling results presented here both indicate the importance of accounting for the deformation of fracture surfaces in calculating changes in fracture aperture with stress. The resulting changes in void geometry, which have important implications for predicting fluid flow through fractured rock, are not predicted by methods that assume that the aperture changes uniformly across the fracture. Again, this points to the importance of coupling measurement and imaging techniques with physical models.

For future work, a technique is being developed that will allow casts to be made while fractures are under a normal load. Aperture maps created from the images of these casts can be compared directly to those obtained from mechanical models, thus providing a way to validate and improve the models. The methods described here are also being extended to analyze the behavior of fractures under shear loads.

FIGURES

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Paper 132, Figure 1.

Figure 1. The variation in intensity across the light source ( la) is evident as a halo on an ~mage of the light source (lb). The fracture images must be corrected for trends introduced by the inhomogeneity of the light source.

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Figure 2. Grey-level images of the cast of a natural fracture before (2a) and after (2b) correction for trends introduced by the light source.

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Figure 3. Image of the calibration wedge before and after correction for trends introduced by the inhomogeneity of the light source.


Figure 4. Images of casts made of two different natural fractures in granite. Laboratory measurements of normal stress versus normal displacement are shown in Figures 7 (for the sample shown in 4a) and 8 (for the sample shown in 4b).

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Figure 5. Two-dimensional cross-section through a fracture (a); represented as a single composite surface (b); and discretized into elements (c). In the model, each bar in (c) is represented by a cylindrical element between elastic half-spaces.

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Figure 6. Schematic diagram of a two-dimensional cross-section through a composite surface of an idealized fracture (a). If all deformation is assumed to be due to asperity deformation then the geometry of the void space remains unchanged under a normal load (b); when deformation of the fracture surfaces is accounted for there are substantial changes in void geometry with increasing load (c).

ISSN 0148-9062

To cite this paper : Int. J. R o c k Mech . & Min. Sci. 34:3-4, p a p e r No . 132. Pub l i shed b y Elsev ie r Sc ience Ltd 1997


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Figure 9. For the fracture pictured in Figure 4a, and an average normal displacement across the fracture of 90 gm, contact area (in white) calculated via image analysis alone (a) and using the analytical model for fracture deformation (b). In 7b, the grey-scale corresponds to the force acting on each element with white corresponding to the highest forces and dark grey corresponding to the lowest.


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Figure 10. For the fracture pictured in Figure 4b, and an average normal displacement across the fracture of 90 gm, contact area (in white) calculated via image analysis alone (a) and using the analytical model for fracture deformation (b). In 8b, the grey-scale corresponds to the force acting on each element with white corresponding to the highest forces and dark grey corresponding to the lowest.

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References

References

Billaux D., Gentler S. 1990. Numerical and laboratory studies of flow in a fracture. In N. Barton and O. Stephansson (eds.), Prec. International Symposium on ReckJeints, Loen, Norway. Balkema, Rotterdam, 369-373.

Gentler S., Billaux D., van Vliet L. 1989. Laboratory testing of the voids of a fracture. Technical Note. Rock Mechanic's and Rock Engineering, 22, 149-157.

Hopkins Deborah L. 1991. The Effect e f Surface Roughness on Joint Stiffness, Aperture, and Seismic Wave Propagation, Ph.D. Dissertation, University of California, Berkeley.

Hopkins D. L., Cook N. G. W., Myer L. R. 1990. Normal joint stiffness as a function of spatial geometry and surface roughness. In N. Barton and O. Stephansson (eds.), Prec. International Symposium on Rock Joints, Loen, Norway. Balkema, Rotterdam, 203-210.

Pyrak-Nolte Laura, Myer Larry R., Cook Neville G. W., Witherspoon Paul A. 1987. Hydraulic and mechanical properties of natural fractures in low permeability rock. In G. Herget and S. Vongpaisal (eds.), Prec. 6th International Congress on Rock Mechanics, Montreal. Balkema, Rotterdam, 224-231.

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