**T**o build predictive models and understand the behavior of complex fracture systems it is necessary to understand the behavior of a single fracture under the same in situ conditions imposed on the whole system. This is true whether the fractures are represented explicitly, as in a network model, or implicitly, as in a stochastic effective medium model.

Two questions are the focus of this chapter: (1) How do flow and transport occur in a single fracture, and what factors significantly affect flow and transport properties? (2) What are the geophysical characteristics of single fractures that make it possible to detect and characterize them remotely?

This chapter emphasizes the results of theoretical and laboratory investigations. Theoretical studies, in combination with controlled laboratory tests, provide the fundamental physical properties and relationships between parameters that describe the processes of fluid flow and transport, seismic wave propagation, and electrical conduction. These fundamental relationships can then be incorporated in models of natural, more complex fracture systems. In laboratory studies, initial and boundary conditions can be controlled and experiments constructed in order to study the effects of a single or limited number of parameters. It should be noted that the physical properties of systems of fractures may be more complex than a simple linear superposition of the properties of single fractures, especially when the fractures intersect. Nonetheless, laboratory experiments can provide the basis for understanding the phenomena that govern large-scale field experiments of fracture systems.

This chapter begins with a discussion of the geometry of a single fracture and the effects of stress and genesis on that geometry. This is followed by a discussion of flow and transport and, finally, of geophysical properties.

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3
Physical Properties and Fundamental Processes in Fractures
To build predictive models and understand the behavior of complex fracture systems it is necessary to understand the behavior of a single fracture under the same in situ conditions imposed on the whole system. This is true whether the fractures are represented explicitly, as in a network model, or implicitly, as in a stochastic effective medium model.
Two questions are the focus of this chapter: (1) How do flow and transport occur in a single fracture, and what factors significantly affect flow and transport properties? (2) What are the geophysical characteristics of single fractures that make it possible to detect and characterize them remotely?
This chapter emphasizes the results of theoretical and laboratory investigations. Theoretical studies, in combination with controlled laboratory tests, provide the fundamental physical properties and relationships between parameters that describe the processes of fluid flow and transport, seismic wave propagation, and electrical conduction. These fundamental relationships can then be incorporated in models of natural, more complex fracture systems. In laboratory studies, initial and boundary conditions can be controlled and experiments constructed in order to study the effects of a single or limited number of parameters. It should be noted that the physical properties of systems of fractures may be more complex than a simple linear superposition of the properties of single fractures, especially when the fractures intersect. Nonetheless, laboratory experiments can provide the basis for understanding the phenomena that govern large-scale field experiments of fracture systems.
This chapter begins with a discussion of the geometry of a single fracture and the effects of stress and genesis on that geometry. This is followed by a discussion of flow and transport and, finally, of geophysical properties.

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GEOMETRIC PROPERTIES AND STRESS EFFECTS
Although fractures are often visualized conceptually as parallel plates separated by some distance (the ''aperture"), there are few situations under natural conditions in which this conceptualization is accurate or meaningful. Generally, in the plane of the fracture there are regions in which the surfaces are in contact and regions, or voids, where the surfaces are not in contact. The factors that most affect the geometry of the void space are (1) the geological origin of the fracture (see Chapter 2); (2) the subsequent changes in stress brought about by natural processes and the activities of humans, such as withdrawal or injection of fluids or construction of underground openings (see Chapter 7); and (3) mineral precipitation and dissolution as fluids flow through a fracture (see Chapter 7).
The voids of a fracture form an interconnected network through which fluid flows. The concept of a fracture as a planar network of interconnected voids leads to an analogy with porous media. In fact, a better conceptualization of a fracture is that of a two-dimensional porous medium in which the fracture voids represent pores. The two-dimensionality arises because all the connections between the fracture voids are in the plane of the fracture.
There is an important difference between fracture void structure and pore structure that limits the extent of this analogy and leads to important differences in hydrological behavior between fractures and porous media. Whereas the bulk of the flow is carried in the large void spaces in both fractures and porous media, the fracture voids tend to be more cracklike in shape compared to the equant shapes of voids in porous media. Cracklike voids deform more easily under applied stress than equant voids. Consequently, the hydrological properties of fractures are subject to a much higher degree of stress sensitivity than porous media. Because fractures represent discontinuities in material properties, shear stresses can also result in deformations, leading to large changes in hydrological properties. Stress sensitivity constitutes the single greatest distinction in hydrological properties between porous media and fractures.
Another consequence of the two-dimensional nature of fractures is that flow can easily become channelized and under two-phase conditions can lead to strong phase interference. Because the pore structure is two-dimensional, flow only has the plane of the fracture to find a path around the closed void (if the matrix rock is impermeable). In porous media (or if the matrix rock is significantly permeable) the third dimension is also available.
One focus of this chapter is the effect of stress on both hydrological and geophysical properties. Fundamental to understanding these effects is a knowledge of how fracture void geometry changes under generalized stress conditions. Thus, the mechanical deformation of fractures is discussed in this section; the effects of this deformation on fluid flow and geophysical properties are discussed in subsequent sections.
The dimensions of the voids in the network certainly affect flow, but of equal if not greater importance is the connectivity of the network. At the most

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fundamental level, flow and transport in a fracture are controlled by the geometric properties of the interconnected voids in the fracture. This section will first discuss these geometric properties from a purely descriptive perspective. Then the changes that occur in geometry as stresses are changed will be described.
Roughness of Fracture Surfaces
A fracture can be envisioned as two rough surfaces in contact. If the surfaces are not perfectly matched, as would be anticipated in many natural environments, there will be some regions where voids remain. Because the void space arises from a mismatch at some scale between the two surfaces of a fracture, a great deal of work has been done to quantify and model the statistics of fracture surfaces.
Surface roughness has been studied extensively because it is an important parameter in the field of tribology. Thomas (1982) gives a complete review of surface roughness measurements. Surface roughness is most commonly measured by profilometry; that is, a sharp stylus is dragged over the surface along a straight line to record the surface height in the form of a profile. For rock fractures, mechanical profilometers (Swan, 1981; Brown et al., 1986; Gentier and Ries, 1990) and optical methods (Miller et al., 1990; Voss and Shotwell, 1990) have been used to measure roughness profiles. The deviation of a surface from its mean plane as determined from these profiles is assumed to be a random process for which statistical parameters such as the variances of height, slope, and curvature are used for characterization. There are a myriad of surface roughness standards that have evolved from different applications. Thomas (1982) describes more than 20 standards that include measures such as the average deviation from the mean (also known as the root-mean-square, or rms, roughness) and peak-to-valley height.
For many rough surfaces, especially those formed in part by natural processes, such as fractures in rock (see Figures 2.3 and 2.26), the rms roughness is a function of the length of the sample (e.g., Sayles and Thomas, 1978; Brown and Scholz, 1985; Power et al., 1987). Consequently, instruments with different resolutions and scan lengths yield different values for these parameters for the same surface. The conventional methods of surface roughness characterization such as rms values are therefore plagued by inconsistencies. The underlying problem with conventional methods is that, although surfaces contain roughness components at a large number of length scales, each roughness measure depends on only a few particular length scales, which are determined by the resolution and dynamic range of the instrument. These difficulties can be overcome when the general nature of surface roughness is understood.
Two key quantities that describe a two-dimensional random process (such as the topography of rough surfaces) are the probability density function for heights and the autocorrelation function (e.g., Whitehouse and Archard, 1970) or an equivalent measure, the power spectrum (Bendat and Piersol, 1971; Bath,

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1974) for texture. The probability density function describes the distribution of surface heights about the mean without regard to horizontal spatial position. The autocorrelation function or power spectrum describes the texture or spatial correlation of heights on the surface. (The power spectrum is the Fourier transform of the autocorrelation function.) This statistical description of the random processes is complete only if the process is Gaussian.
Brown and Scholz (1985) and Power et al. (1987) computed the power spectral density, G(k), for various rock surfaces, natural joint surfaces in crystalline and sedimentary rocks, a bedding-plane surface, and frictional wear surfaces. Their results show that there is remarkable similarity among these surfaces. Profiles of these widely different surfaces yield power law power spectra of the form
where k is the wavenumber related to the wavelength of surface roughness according to k = 2π/. The exponent has a fairly limited range (typically between -2 and -3). The power spectrum (and therefore the roughness) can thus be described to first approximation by two parameters: (1) the exponent (i.e., the slope of the power spectrum on a log-log plot) and (2) the proportionality constant B (i.e., the value of G(k) at k = 1).
This power law form of the power spectrum indicates that fracture surface topography can be represented in terms of fractal geometry where the fractal dimension of the surface, D, is related to the power spectrum exponent as D = (7 - )/2 (see discussions by Mandelbrot, 1982; Brown and Scholz, 1985; Brown, 1987; and Power and Tullis, 1991). This relationship is strictly true only if the phase spectrum is random and the topographic heights have a normal distribution. The fractal dimension describes the proportion of high-frequency to low-frequency roughness elements and is a measure of surface texture. For natural fracture surfaces, D falls in the range of 2 to 3, with small values representing smoother surfaces. Variations in over different measurement scales indicate, however, that it may be unrealistic to extrapolate the power spectrum outside the range of measured wavelengths.
Void Geometry
Current understanding of the geometric properties of fracture void space has evolved from both experimental and theoretical investigations. These investigations have addressed two main issues: (1) What are the geometric characteristics at some nominal stress? (2) How do these characteristics change as stress changes? In the following discussion the term local aperture refers to the distance between the two opposing fracture surfaces measured perpendicular to the plane of the fracture.

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Various techniques, each with advantages and disadvantages, have been used to characterize fracture void geometry. Representative examples of techniques described here are grouped into two categories: (1) those that use measurements of the roughness of the two surfaces composing the fracture and (2) those that involve filling the void space with a casting material. In the first category one method is to sum the surface roughness heights from both sides of the fracture to generate a "composite topography." Specifically, if the heights of the surfaces on opposing walls of the fracture are h1(x,y) and h2(x,y), measured relative to the mean level of each surface with positive values increasing outward from the surface, the composite topography is defined as hc(x,y) = h1(x,y) + h2(x,y). The composite topography contains only the mismatched parts of the surface roughness and represents the distribution of local apertures under essentially zero stress conditions (Figure 3.1). In a more sophisticated approach to the use of
FIGURE 3.1 Example of measured matched profiles and corresponding composite topography. Notice that the high-amplitude long-wavelength components present in the profiles of the individual surfaces are not present in the composite topography. From Brown et al. (1986).

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measurements of fracture surface roughness, Roberds et al. (1990) showed that the statistical properties of local apertures can be derived directly from the statistical properties of each of the opposing fracture surfaces. The advantage of methods that use surface roughness data is that such data can be easily obtained. The disadvantages are that changes from nonzero stress conditions must be inferred from models and it may be difficult to match the two halves of the fracture correctly.
In the second category are methods that involve filling the void space with different materials such as epoxy (Gale, 1987; Billaux and Gentier, 1990) or Wood's metal (Pyrak-Nolte et al., 1987a) and other casting techniques (e.g., Hakami and Barton, 1990). Epoxy castings provide good estimates of local aperture geometry, but some techniques require that the sample be sliced and consequently destroyed. Estimates of the areal distribution of void space can be obtained accurately by using Wood's metal, but it is difficult to obtain local aperture geometry. Both local aperture and void distribution can be obtained from other casting methods, but there are problems with resolution.
Although laboratory techniques have shortcomings, a number of insights concerning fracture void geometry have been provided by such measurements. Brown et al. (1986) measured matched profiles from both halves of natural joint surfaces. The power spectrum of local apertures and the composite topography were computed as well. They showed that the local aperture distribution has a power law spectrum at high spatial frequencies (short wavelengths) but flattens out at low spatial frequencies (long wavelengths), as illustrated in Figure 3.2. The crossover between the power law behavior and the flat spectrum allows a mismatch length scale to be defined as the point where the ratio of the local aperture spectrum to the spectrum of the individual surfaces reaches one-half its high-frequency value (Figure 3.2). In the simplest model this mismatch length scale is the third parameter needed to define the roughness of a fracture. This scale is essentially the largest wavelength present in the composite topography with any significant amplitude and defines the dominant roughness component of the fracture.
The results of Wood's metal injection tests on two natural granitic fractures subjected to a normal stress of 33 MPa are shown in Figure 3.3. The figure illustrates the inherent heterogeneity in the areal distribution of the void space in natural fractures. Though heterogeneous, it is also clear that the void space is spatially correlated. Nolte et al. (1989) studied the fractal properties of the void geometry of natural granitic fractures, finding that the fractal dimension, D, ranged from about 1.99 to 1.95, depending on the stress level. Further studies (Pyrak-Nolte et al., 1992) suggest that multifractal analyses of fracture void space may distinguish between successive processes that might have altered the fracture void structure.
Measurements (Gale, 1990) of an epoxy cast of a granite fracture indicate that local apertures sizes are lognormally distributed. There is an apparent relationship

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FIGURE 3.2 (a) Power spectra of the surface profiles and aperture (composite topography) shown in Figure 3.1. Notice the lack of power at low frequencies in the aperture relative to the surfaces. (b) Ratio of the aperture spectrum to the surface spectrum. A mismatch length scale c can be defined by the spatial frequency at which the ratio falls to one-half its high-frequency asymptotic value. For length scales greater than c the surfaces are closely matched, and for length scales less than c the surfaces are mismatched. From Brown et al. (1986).

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FIGURE 3.3 Composite SEM micrograph of Wood's metal casts of two natural fractures showing void space at 33 MPa effective stress. Void space, filled by Wood's metal, is shown in white; contact areas are black. From Pyrak-Nolte et al. (1992).

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between local aperture size and void extent. That is, large local apertures are associated with laterally extensive voids, whereas small local apertures are associated with restricted voids.
Inferences about the statistical properties of fracture void space have been tested in a number of modeling studies. Several approaches have yielded aperture distributions that visually have properties similar to the experimental results shown in Figure 3.3. Brown (1987), Thompson and Brown (1991), and others have used the fractal properties of surface roughness measurements in combination with the mismatch length scale to generate such patterns. Moreno et al. (1988), Tsang and Tsang (1987), and others have created spatially correlated local aperture patterns, such as those shown in Figure 3.3, by using statistical techniques that assumed lognormal local aperture size distributions. Pyrak-Nolte et al. (1988) used a stratified percolation model to create spatially correlated local aperture patterns that are approximately lognormal. Although lognormal local aperture size distributions are most commonly used in models, analysis of stress/deformation data by Hopkins et al. (1990) indicate that some local aperture distributions may be bimodal.
STRESS EFFECTS ON FRACTURE VOID GEOMETRY
Deformation in a fracture can arise from either a change in fluid pressure or a perturbation of the stress field in the rock. The important stress for mechanical behavior and fluid flow in fractures is the effective stress, which is generally taken to be the normal stress on the fracture minus the fluid pressure (Terzaghi, 1936). In a three-dimensional stress field the normal stress is the component of the total stress measured perpendicular to the fracture plane. By convention in rock mechanics, compressive stresses are taken to be positive in sign, so effective stress values are usually positive. However, in some instances, such as in a hydrofracture, the fluid pressure exceeds the far-field normal stress, so effective stress values are negative.
Under transient conditions, effective stress conditions may be different in fractures and the surrounding porous media. If the matrix permeability is different than the fracture permeability, fluid pressures will be different under transient flow conditions. If matrix permeability is less than the fracture permeability, there will be a larger effective stress in the fracture than in the matrix when fluid is withdrawn. Differences in effective stresses will also arise when the pore compressibility of the rock matrix is greater than the void compressibility of the fractures. In this case, equal reductions in pore and fracture pressures will produce larger effective stress changes in the fracture than in the rock.
When the stress on a fracture changes, it will deform. Because of the importance of fractures to the mechanical behavior of rock masses, a large number of experimental investigations have addressed deformability of fractures. A good summary is presented by Einstein and Dowding (1981). The results of their work

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have been incorporated into models of rock mass mechanical behavior (e.g., Roberds and Einstein, 1978). However, the following discussion is restricted to deformability from the standpoint of how it affects the process of fluid flow.
The stress on a fracture can be decomposed into two components, one in a direction perpendicular to the surface (normal) and one in a direction parallel to the surface (shear). In general, the effects of these two components are highly coupled; that is, the deformation caused by a change in one component is dependent on the magnitude of the other. Fracture surface roughness is one of the primary reasons for this coupling, as illustrated in Figure 3.4. This figure shows the distribution of voids and contacting asperities in an idealized representation of a very rough undeformed fracture. The application of shear stress to this fracture at a very low normal stress may cause one surface to ride up and over the asperities of the other, leading to large dilation. At the other extreme, at very high normal stress, the frictional forces resisting slip may exceed the strength of the rock and the asperities will be sheared off. Dilation would be minimal in this case. Deformation associated with these loading conditions is discussed later in this chapter under "Shear Stress Effects." Considerable work, as discussed below, has been carried out for the case in which only changes in effective normal stress are considered. Referring again to Figure 3.4, such studies address closure of voids where the surfaces are not in contact and deformation of asperities where the surfaces are in contact.
Normal Stress Effects
When normal stress is applied to a natural fracture in the laboratory, fracture deformation is typically nonlinear (Figure 3.5). The rate of deformation is greatest at low values of normal stress, indicating that fracture stiffness increases as normal stress increases.
A common feature of fracture deformation is a hysteresis effect during stress loading and unloading, as shown Figure 3.5. In hard clean crystalline rock fractures, hysteresis is almost entirely due to processes arising from mismatch
FIGURE 3.4 Cross section through a very rough fracture in the underformed state showing distribution of voids and contacting asperities.

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FIGURE 3.5 Measurements of the closure of natural joints under normal stress (n). From Bandis et al. (1981).

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APPENDIX 3.C
INFLUENCE OF TWO-PHASE STRUCTURE ON FRACTURE PERMEABILITY AND SOLUTE TRANSPORT
Under steady-state conditions, flow in a single-phase along the plane of a fracture is confined to the connected network of apertures filled by that phase. Decreasing phase saturation reduces the fraction of the fracture plane available for flow. The resultant decrease in cross-sectional area and increased flow tortuosity reduces the relative permeability for the phase.
Nicholl and Glass (1994) used a transparent analog rough-walled fracture to explore the effects of phase structure on fluid permeability and solute dispersion. Flow through the connected phase of a two-phase structure exhibited a channelization that was not observed under single-phase conditions (Figures 3.C1, 3.C2, and 3.C3). Path tortuosity of the channelized flow was observed to increase with complexity of the phase structure and size of the entrapped regions. Disconnected regions filled by the flowing phase are entirely isolated from flow under steady-state conditions. In addition, even in the connected regions, highly tortuous flow paths tend to bypass significant portions of the connected phase structure. This creates dead zones that do not actively participate in flow but communicate with active regions through diffusion. As a result, the average cross-sectional area of the connected phase structure is larger than the effective flow area.
For phase structures formed under nonequilibrium conditions, aperture filling may not follow a simple pressure/size relationship (Glass, 1993), further complicating the phase structure/permeability relationship. The fluid phase shown in Figure 3.C1 occupies 81.2 percent of the fracture plane. However, fluid permeability was only 14.3 percent of that measured at 100 percent saturation.
As expected from the previous discussion, two-phase wetted structure significantly affects solute dispersion. Flow channelization creates differential advection in the plane of the fracture (Figures 3.C2 and 3.C3). Velocity differentials introduced by local variations in cross-sectional area also act to increase dispersion. Dead zones created by flow tortuosity act to significantly extend the tail of the residence time distribution curve.

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FIGURE 3.C1 Steady-state wetted structure in a transparent analog fracture. Dark areas are filled with dyed water; light regions are entrapped air. From Nicholl and Glass (1994).

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FIGURE 3.C2 Tracer pulse consisting of clear water entering the two-phase wetted structure in Figure 3.C1 from a constant flow boundary. From Nicholl and Glass (1994).

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FIGURE 3.C3 Further development of the tracer pulse shown in Figure 3.C2. Note wholly isolated regions and dead zones that communicate by diffusion. From Nicholl and Glass (1994).

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REFERENCES
Adamson, A. W. 1990. Physical Chemistry of Surfaces, 5th ed. New York: John Wiley & Sons, 777 pp.
Angel, Y. C., and J. D. Achenbach. 1985. Reflection and transmission of elastic waves by a periodic array of cracks. Journal of Applied Mechanics, 52:33–41.
Archambault, G., M. Fortin, D. E. Gill, M. Aubertin, and B. Ladanzi. 1990. Experimental investigations for an algorithm simulating the effect of variable normal stiffness on discontinuities shear strength. Pp. 141–148 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton, and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Archie, G. E. 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the American Institute of Mechanical Engineering, 146:54–62.
Bandis, S. C. 1990. Mechanical properties of rock joints. Pp. 125–140 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Bandis, S., A. C. Lumsden, and N. R. Barton. 1981. Experimental studies of scale effects on the shear behavior of rock joints. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts, 18:1–21.
Barton, N. 1973. Review of a new shear-strength criterion for rock joints. Engineering Geology, 7:287–332.
Barton, N., S. Bandis, and K. Bakhtor. 1985. Strength, deformation and conductivity coupling in rock joints. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 22(3):121–140.
Bath, M. 1974. Spectral Analysis in Geophysics. New York: Elsevier.
Bendat, J. S., and A. G. Piersol. 1971. Random Data, Analysis and Measurement Procedures. New York: John Wiley & Sons.
Bernabe, Y. 1986. Pore volume and transport properties changes during pressure cycling of several crystalline rocks. Mechanics and Materials, 5:235–249.
Bernabe, Y. 1988. Comparison of the effective pressure law for permeability and resistivity formation factor in Chelmsford granite. Pure and Applied Geophysics, 127:607–625.
Billaux, D., and S. Gentier. 1990. Numerical and laboratory studies of flow in a fracture. Pp. 369–373 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Biot, M. A. 1956. Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range. Journal of the Acoustics Society of America, 28(2):168.
Brace, W. F., A. S. Orange, and T. R. Madden. 1965. Effect of pressure on the electrical resistivity of water-saturated crystalline rocks. Journal of Geophysical Research, 70:5669–5678.
Broadbent, S. R., and J. M. Hammersley. 1957. Percolation Processes. I. Crystals and Mazes. Cambridge Philosophical Society Proceedings, 53:629–641.
Brown, S. R. 1987. Fluid flow through rock joints: the effect of surface roughness. Journal of Geophysical Research, 92(82):1337–1347.
Brown, S. R. 1989. Transport of fluid and electric current through a single fracture. Journal of Geophysical Research, 94(B7):9429–9438.
Brown, S. R., and C. H. Scholz. 1985. Broad bandwidth study of the topography of natural rock surfaces. Journal of Geophysical Research, 90:575–582.
Brown, S. R., and C. H. Scholz. 1986. Closure of rock joints. Journal of Geophysical Research, 91:4939–4948.
Brown, S. R., R. L. Kranz, and B. P. Bonner. 1986. Correlation between the surfaces of natural rock joints. Geophysics Research Letter, 13:1430–1433.
Budiansky, B., and R. J. O'Connell. 1980. Bulk dissipation in heterogeneous media. Solid Earth Geophysics and Geotechnology, New York: American Society of Mechanical Engineers, 42:1.

OCR for page 103

Rock Fractures and Fluid Flow: Contemporary Understanding and Applications
Carlson, R. L., and A. F. Gangi. 1985. Effect of cracks on the pressure dependence of P-wave velocities in crystalline rocks. Journal of Geophysical Research, 90(B10):8675–8684.
Carter, J. J. 1990. The effects of clay coating on fluid flow through simulated rock fractures. M.S. thesis, University of California, Berkeley.
Chouke, R. L., P. van Meurs, and C. van der Poel. 1959. The instability of slow immiscible, viscous liquid-liquid displacements in porous media. Transactions of the American Institute of Mining Engineers, 216:T.P. 8073.
Cook, A. M., L. R. Myer, N. G. W. Cook, and F. M. Doyle. 1990. The effect of tortuosity on flow through a natural fracture. Pp. 371–378 in Rock Mechanics Contributions and Challenges, Proceedings of the 31st U.S. Symposium on Rock Mechanics, W. A. Hustrulid and G. A. Johnson, eds. Rotterdam: A. A. Balkema.
David, C., Y. Gueguen, and G. Pampoukis. 1990. Effective medium theory and network theory applied to the transport properties of rock. Journal of Geophysical Research, 95(B5):6993–7005.
de Gennes, P. G. 1985. Wetting: statics and dynamics. Reviews of Modern Physics, 57:827–863.
Doyen, P. M. 1988. Permeability, conductivity, and pore geometry of sandstone. Journal of Geophysical Research, 93(B7):7729–7740.
Dussan, V. 1979. On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annual Review of Fluid Mechanics, 11:371–400.
Einstein, H. H., and C. H. Dowding. 1981. Shear resistance and deformability of rock discontinuities. Pp. 177–219 in Physical Properties of Rocks and Minerals, Y. S. W. Touloukian, R. Judd, and R. E. Ray, eds. New York: McGraw-Hill.
Esaki, T., H. Hojo, T. Kimura, and N. Kameda. 1991. Shear-flow coupling test on rock joints. Pp. 389–392 in Proceedings of 7th International Congress on Rock Mechanics. Rotterdam: A. A. Balkema.
Feder, J. 1988. Fractals. New York: Plenum Press, 283 pp.
Gale, J. E. 1982. The effects of fracture type (induced versus natural) on the stress-fracture closure fracture permeability relationships. Pp. 290–298 in Issues in Rock Mechanics, Proceedings of the 23rd U.S. Symposium on Rock Mechanics, R. Goodman and F. Henze, eds. New York: Society of Mining Engineers of AIME.
Gale, J. E. 1987. Comparison of coupled fracture deformation and fluid flow models with direct measurements of fracture pore structure and stress-flow properties. Pp. 1213–1222 in Rock Mechanics, Proceedings of the 28th U.S. Symposium on Rock Mechanics, I. W. Farmer, J. J. K. Daemen, C. S. Desai, C. E. Glass, and S. P. Neuman, eds. Rotterdam: A. A. Balkema.
Gale, J. 1990. Hydraulic behavior of rock joints. Pp. 351–362 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Gale, J. E., and E. J. Reardon. 1984. Effect of groundwater geochemistry on the permeability of grouted fractures. Canadian Geotechnical Journal, 21(1):8–20.
Gentier, S., and J. Ries. 1990. Quantitative description and modeling of joint morphology. Pp. 375–382 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Glass, R. J. 1993. Modeling gravity-driven fingering in rough-walled fractures using modified percolation theory. Pp. 2042–2052 in Proceedings of the Fourth Annual International High-Level Radioactive Waste Management Conference. La Grange Park, Ill.: American Nuclear Society.
Glass, R. J., and D. L. Norton. 1992. Wetted region structure in horizontal unsaturated fractures: water entry through the surrounding porous matrix. Pp. 717–726 in Proceedings of the Third Annual International High-Level Radioactive Waste Management Conference. La Grange Park, Ill.: American Nuclear Society.
Goodman, R. E. 1976. Methods of Geological Engineering Discontinuous Rock. St. Paul: West Publishing Co.

OCR for page 103

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Hakami, E., and N. Barton. 1990. Aperture measurements and flow experiments using transparent replicas of rock joints. Pp. 383–390 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Heard, H. C., and L. Page. 1982. Elastic moduli, thermal expansion, and inferred permeability of two granites to 350°C and 55 megapascals. Journal of Geophysical Research, 87(B11):9340–9348.
Homsy, G. M. 1987. Viscous fingering in porous media. Annual Review of Fluid Mechanics , 19:271–311.
Hopkins, D. C., N. G. W. Cook, and L. R. Myer. 1990. Normal joint stiffness as a function of spatial geometry and surface roughness. Pp. 203–210 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Hudson, J. A. 1980. Overall properties of a cracked solid. Mathematical Proceedings of the Cambridge Philosophical Society, 88:371.
Iwai, K. 1976. Fundamental studies of fluid flow through a single fracture. Ph.D. thesis, University of California, Berkeley.
Jones, F. O. 1975. A laboratory study of the effects of confining pressure on fracture flow and storage capacity in carbonate rocks. Journal of Petroleum Technology, Jan., pp. 21–27.
Jones, T. D. 1986. Pore fluids and frequency-dependent wave propagation in rocks. Geophysics, 51(10):1939–1953.
Khilar, K. L., and H. S. Folger. 1984. The existence of a critical salt concentration for particle release. Journal of Colloid and Interface Science, 101(1):214–224.
King, M. S., L. R. Myer, and J. J. Rezowalli. 1986. Experimental studies of elastic-wave propagation in a columnar-pointed rock mass. Geophysical Prospecting, 34(8):1185–1199.
Kranz, R. L., A. D. Frankel, T. Engelder, and C. H. Scholz. 1979. The permeability of whole and jointed Barre granite. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts, 16:225–234.
Kutter, H. K., and F. Otto. 1990. Influence of parallel and cross joints on shear behavior of rock discontinuities. Pp. 243–250 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Lamb, H. 1932. Hydrodynamics, 6th ed. Cambridge: Cambridge University Press.
Lehner, F. K., and J. Bataille. 1984. Nonequilibrium thermodynamics of pressure solution. Report No. 25, Division of Engineering, Brown University, Providence, R.I., 52 pp.
Magnusson, K. A., S. Carlsten, and O. Olsson. 1987. Crosshole investigations—physical properties of core samples from boreholes F1 and F2. Stripa Project IR 87-10. Stockholm: Swedish Nuclear Fuel and Waste Management Company (SKB).
Makurat, A. 1985. The effect of shear displacement on the permeability of natural rough joints. Pp. 99–106 in Proceedings Hydrogeology of Rocks of Low Permeability, Tucson, Ariz., International Association of Hydrogeologists, Memoires XVII .
Mandelbrot, B. B. 1982. The Fractal Geometry of Nature. San Francisco: W. H. Freeman and Co.
Miller, S. M., P. C. McWilliams, and J. C. Kerkering. 1990. Ambiguities in estimating fractal dimensions of rock fracture. Pp. 471–478 in Rock Mechanics Contributions and Challenges, Proceedings of the 31th U.S. Symposium on Rock Mechanics, W. A. Hustrulid and G. A. Johnson, eds. Rotterdam: A. A. Balkema.
Mindlin, R. D. 1949. Compliance of elastic bodies in contact, Journal of Applied Mechanics, 16:259–269.
Mindlin, R. D., and H. Deresiewicz. 1953. Elastic spheres in contact under varying oblique forces. Journal of Applied Mechanics, 20:327–344.
Mitchell, S. K. 1993. Fundamentals of Foil Behavior, 2d ed. New York: John Wiley & Sons.
Moreno, L., Y. W. Tsang, C. F. Tsang, F. V. Hale, and I. Neretnieks. 1988. Flow and transport in a single fracture: a stochastic model and its relation to some field observations. Water Resources Research, 24(12):2033–2048.

OCR for page 103

Rock Fractures and Fluid Flow: Contemporary Understanding and Applications
Myer, L. 1991. Hydromechanical and seismic properties of fractures. Pp. 397–409 in Proceedings of the 7th International Rock Mechanics Congress, vol. 1
Myer, L. R., D. Hopkins, and N. G. W. Cook. 1985. Effects of contact area of an interface on acoustic wave transmission. Pp. 549–556 in Research and Engineering Applications in Rock Mechanics, Proceedings of the 26th U.S. Symposium on Rock Mechanics. Rotterdam: A. A. Balkema.
Myer, L. R., L. S. Pyrak-Nolte, and N. G. W. Cook. 1990a. Effect of single fractures on seismic wave propagation. Pp. 476–473 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Myer, L. R., L. J. Pyrak-Nolte, D. Hopkins, and N. G. W. Cook. 1990b. Seismic characterization of fracture properties. Pp. 908–914 in Proceedings of First Annual International High-Level Radioactive Waste Management Conference, vol. 1. La Grange Park, Ill.: American Nuclear Society.
Nelson, P. H., R. Rachiele, and J. S. Remer. 1981. Water inflow into boreholes during the Stripa Heater Experiments. LBL-12574, Lawrence Berkeley Laboratory, Berkeley, Calif.
Nicholl, M. J., and R. J. Glass. 1994. Wetting phase permeability in a partially saturated horizontal fracture. Pp. 2007–2019 in Proceedings of the 5th International Conference on High-Level Radioactive Waste Management. La Grange Park, Ill.: American Nuclear Society.
Nicholl, M. J., R. J. Glass, and H. A. Nguyen. 1992. Gravity-driven fingering in unsaturated fractures. Pp. 321–332 in High-Level Radioactive Waste Management, Proceedings of the Third Annual International Conference, vol. 1. La Grange Park, Ill.: American Nuclear Society.
Nicholl, M. J., R. J. Glass, and H. A. Nguyen. 1993a. Wetting front instability in an initially wet unsaturated fracture. Pp. 2061–2070 in Proceedings of the Fourth Annual International High-Level Radioactive Waste Management Conference, vol. 2. La Grange Park, Ill.: American Nuclear Society.
Nicholl, M. J., R. J. Glass, and H. A. Nguyen. 1993b. Small scale behavior of single gravity-driven fingers in an initially dry fracture. Pp. 2023–2032 in Proceedings of the Fourth Annual International High-Level Radioactive Waste Management Conference, vol. 2. La Grange Park, Ill.: American Nuclear Society.
Nolte, D. D., L. J. Pyrak-Nolte, and N. G. W. Cook. 1989. The fractal geometry of flow paths in natural fractures in rock and the approach to percolation. PAGEOPH, 131(1/2):271.
Noorishad, J., C. F. Tsang, and P. A. Witherspoon. 1984. Coupled thermal-hydraulic-mechanical phenomena in saturated fractured porous rocks: numerical approach. Journal of Geophysical Research, 89(B12):10365–10375.
Ohnishi, Y., and P. G. R. Dharmaratra. 1990. Shear behavior of physical models of rock joints under constant normal stiffness conditions. Pp. 267–273 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Olsson, W. A. 1992. The effect of slip on the flow of fluid through a fracture. Geophysics Research Letters, 19(6):541–543.
Paillet, F. L., A. E. Hess, C. H. Cheng, and E. Hardin. 1987. Characterization of fracture permeability with high-resolution vertical flow measurements during borehole pumping. Ground Water, 25(1):28–40.
Papaliangas, T., A. C. Lumsden, S. A. Hencher, and S. Manolopoulou. 1990. Shear strength of modeled filled rock joints. Pp. 275–282 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema .
Paterson, M. S. 1983. The equivalent channel model for permeability and resistivity in fluid-saturated rocks—a reappraisal. Mechanics and Materials, 2(4):345–352.
Patton, D. F. 1966. Multiple modes of stress failure in rock. P. 509 in Proceedings of the First International Congress on Rock Mechanics, vol. 1. Lisbon: Bertrand.

OCR for page 103

Rock Fractures and Fluid Flow: Contemporary Understanding and Applications
Persoff, P., K. Pruess, and L. Myer. 1991. Two-phase flow visualization and relative permeability measurement in transparent replicas of rough-walled rock fracture. Lawrence Berkeley Laboratory Report 30161, presented at the 16th Workshop on Geothermal Reservoir, Engineering, Stanford, Calif.
Peterson, J. E., Jr., D. Hopkins, and L. Myer. 1993. Application of seismic displacement discontinuity theory to Hanford data. Pp. 1006–1009 in Expanded Abstracts with Biographies, 1993 Technical Program: 63d Annual Meeting and International Exhibition, Society of Exploration Geophysicists.
Piggot, A. R., and D. Elsworth. 1990. Laboratory studies of transport within a single rock fracture. Pp. 397–404 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Power, W. L., and T. E. Tullis. 1991. Euclidean and fractal models for the description of rock surface roughness. Journal of Geophysical Research , 96(B1):415–424.
Power, W. L., T. E. Tullis, S. R. Brown, G. N. Boitnott, and C. H. Scholz. 1987. Roughness of natural fault surfaces. Geophysics Research Letters, 14:29–32.
Pruess, K., and Y. W. Tsang. 1990. On two-phase relative permeability and capillary pressure of rough-walled rock fractures. Water Resources Research, 26(9):1915–1926.
Pyrak-Nolte, L. J., L. R. Myer, N. G. W. Cook, and R. A. Witherspoon. 1987a. Hydraulic and mechanical properties of natural fractures in low permeability rock. Pp. 225–232 in Proceedings of 6th International Congress of Rock Mechanics, vol. I. Rotterdam: A. A. Balkema.
Pyrak-Nolte, L. J., L. R. Myer, and N. G. W. Cook. 1987b. Seismic visibility of fractures. Pp. 47–56 in Rock Mechanics, Proceedings of the 28th U.S. Symposium on Rock Mechanics. Rotterdam: A. A. Balkema.
Pyrak-Nolte L. J., N. G. W. Cook, and D. D. Nolte. 1988. Fluid percolation through single fractures. Geophysics Research Letters, 15(11):1247–1250.
Pyrak-Nolte, L. J., N. G. W. Cook, and L. R. Myer. 1990a. Stratified percolation model for saturated and unsaturated flow through natural fractures. Pp. 551–558 in Proceedings of First Annual International High-Level Radioactive Waste Management Conference, vol. 1. La Grange, Ill.: American Nuclear Society.
Pyrak-Nolte, L. J., L. R. Myer, and N. G. W. Cook. 1990b. Transmission of seismic waves across single fractures. Journal of Geophysical Research, 95(B6):8617–8638.
Pyrak-Nolte, L. J., L. R. Myer, and N. G. W. Cook. 1990c. Anisotropy in seismic velocities and amplitudes from multiple parallel fractures. Journal of Geophysical Research, 95(B7):11345–11358.
Pyrak-Nolte, L. J., L. R. Myer, and D. D. Nolte. 1992. Fractures: finite-size sealing and multifractals. PAGEOPH, 131(1/2):271.
Rathore, J. S., F. Fjaer, R. M. Holt, and L. Reulie. 1991. Acoustic anisotropy of synthetics with controlled crack geometrics. European Association of Exploratory Geophysicists Convention Abstracts, 53:538–539.
Raven, K. G., and J. E. Gale. 1985. Water flow in a natural rock fracture as a function of stress and sample size. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts, 22(4):251–261.
Roberds, W. J., and H. H. Einstein. 1978. Comprehensive model for rock discontinuities. Journal of the Geotechnical Engineering Division, 104(GT5):553–569.
Roberds, W. J., M. Iwano, and H. H. Einstein. 1990. Probabilistic mapping of rock joint surfaces. Pp. 681–691 in Rock Joints, Proceedings of the International Symposium on Rock Joints, Loen, Norway, W. Barton and E. Stephansson, eds. Rotterdam: A. A. Balkema.
Rutter, E. H. 1983. Pressure solution in nature, theory and experiment. Journal of the Geological Society (London), 140:725–740.
Saffman, P. G. 1986. Viscous fingering in Hele-Shaw cells. Journal of Fluid Mechanics, 173:73–94.

OCR for page 103

Rock Fractures and Fluid Flow: Contemporary Understanding and Applications
Saffman, P. G., and G. I. Taylor. 1958. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proceedings of the Royal Society of London, A245:312–331.
Sayles, R. S., and T. R. Thomas. 1978. Surface topography as a non-stationary random process. Nature, 271:431–434.
Schoenberg, M. 1980. Elastic wave behavior across linear slip interfaces. Journal of the Acoustical Society of America, 68(5):1516–1521.
Schoenberg, M. 1983. Reflection of elastic waves from periodically stratified media with interfacial slip. Geophysical Prospecting, 31:265–292.
Schoenberg, M., and F. Muir. 1989. A calculus for finely layered anisotropic media. Geophysics, 54(5):581–589.
Sen, P. N., C. Scala, and M. H. Cohen. 1981. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics, 46:781–795.
Shen, L. C., W. C. Savre, J. M. Price, and K. Ahtavale. 1985. Dielectric properties of reservoir rocks at ultra-high frequencies. Geophysics, 50:692–704.
Sihvola, A. H. 1989. Self-consistency aspects of dielectric mixing theories. IEEE Transactions on Geoscience and Remote Sensing, 27:403–415.
Simandoux, P. 1963. Mesures dielectrique en milleux poreux, application a mesure des saturations en eaux, etude du comportement des massifs argileux. Rev. de l'institut Francais du Petrole, (Suppl.).
Smith, L., C. W. Mase, and F. W. Schwartz. 1987. Estimation of fracture aperture using hydraulic and tracer tests. Pp. 453–463 in Rock Mechanics, Proceedings of the 28th U.S. Symposium on Rock Mechanics, I. W. Farmer, J. J. K. Daemen, C. S. Desai, C. E. Glass, and S. P. Neuman, eds. Rotterdam: A. A. Balkema.
Sneddon, I. N., and A. A. Elliott. 1946. The opening of a griffith crack under internal pressure. Quarterly of Applied Mathematics, IV:262.
Sprunt, E. S., and A. Nur. 1977. Destruction of porosity through pressure solution. Geophysics, 42:726–741.
Stockman, H. W., C. T. Stockman, and C. R. Carrigan. 1990. Modeling viscous segregation in immiscible fluids via lattice gas automata. Nature, 348:523.
Suarez-Rivera, F. R., N. G. W. Cook, and L. R. Myer. 1992. Study of the transmission of shear waves across thin liquid films and thin clay layers. Pp. 937–946 in Rock Mechanics, Proceedings of the 33rd U.S. Rock Mechanics Symposium, J. R. Tillerson and W. R. Warvenish, eds. Rotterdam: A. A. Balkema.
Swan, G. 1981. Tribology and the characterization of rock joints. Pp. 402–407 in Proceedings of the 22nd U.S. Symposium on Rock Mechanics. Cambridge: Massachusetts Institute of Technology.
Tada, H. P., P. C. Paris, and G. K. Irwin. 1973. The Stress Analysis of Cracks Handbook. Helbertown, Pa.: Del Research Corp.
Tang, X. M., C. H. Cheng, and F. L. Paillet. 1991. Modeling borehole Stoneley wave propagation a permeable in-situ fracture. Pp. GG1–GG25 in Society of Professional Well Log Analysts Annual Logging Symposium, 32nd, Transactions. Houston, Tex.: Society of Professional Well Log Analysts.
Terzaghi, K. 1936. The shearing resistance of saturated soils. Pp. 54–56 in Proceedings of the International Conference on Foundation Engineering, Graduate School of Engineering, Harvard University. Boston: Spaulding Moss Co.
Thomas, T. R. 1982. Rough Surfaces. New York: Longman.
Thompson, M. E., and S. R. Brown. 1991. The effect of anisotropic surface roughness on flow and transport in fractures. Journal of Geophysical Research, 96(B13):21923–21932.
Thomsen, L. 1985. Biot-consistent elastic modeli of porous rocks: low-frequency limit. Geophysics, 50(12):2797–2807.
Thomsen, L. 1991. Elastic anisotropy due to aligned cracks in porous rock. European Association of Exploratory Geophysicists Convention Abstracts, 53:244–245.

OCR for page 103

Rock Fractures and Fluid Flow: Contemporary Understanding and Applications
Timoshenko, S. P., and J. N. Goodier. 1970. Theory of Elasticity, 3d ed. New York: McGraw-Hill.
Trimmer, D., B. Bonner, H. C. Heard, and A. Duba. 1980. Effect of pressure and stress on water transport in intact and fractured gabbro and granite. Journal of Geophysical Research, 85(B12):7059–7071.
Tsang, Y. W. 1984. The effect of tortuosity on fluid flow through a single fracture. Water Resources Research, 20:1209–1215.
Tsang, Y. W., and C. F. Tsang. 1987. Channel model of flow through fractured media. Water Resources Research, 23(3):467–479.
Tsang, Y. W., C. F. Tsang, I. Neretnieks, and L. Moreno. 1988. Flow and tracer transport in fractured media: a variable aperture channel model and its properties. Water Resources Research, 24(12):2049–2060.
Voss, C. F., and L. R. Shotwell. 1990. An investigation of the mechanical and hydraulic behavior of tuff fractures under saturated conditions. Pp. 825–834 in High-Level Radioactive Waste Management. La Grange Park, Ill.: American Nuclear Society.
Walsh, J. B. 1981. Effect of pore pressure and confining pressure on fracture permeability. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts, 18:429–435.
Walsh, J. B., and W. F. Brace. 1984. The effect of pressure on porosity and the transport properties of rock. Journal of Geophysical Research, 89:9425–9431.
Wang, J. S. Y., T. N. Narasimhan, and C. H. Scholz. 1988. Aperture correlation of a fractal fracture. Journal of Geophysical Research, 93(B3):2216–2224.
Waxman, M. H., and L. J. M. Smits. 1968. Electrical conductivities of oil-bearing shaley sands. Transactions of the Society of Petroleum Engineers, 243:107–115.
Whitehouse, D. J., and J. F. Archard. 1970. The properties of random surfaces of significance in their contact. Proceedings of the Royal Society of London , Series A, 316:97–121.
Wilkinson, D., and J. F. Willemsen. 1983. Invasion percolation: a new form of percolation theory. Journal of Physics A: Math. Gen., 16:3365–3376.
Witherspoon, P. A., J. S. Y. Wang, K. Iwai, and J. E. Gale. 1980. Validity of cubic law for fluid flow in a deformable rock fracture. Water Resources Research, 16:1016.
Yang, G., N. G. W. Cook, and L. R. Myer. 1989. Network modeling of flow in natural fractures. Pp. 57–64 in Proceedings of the 30th U.S. Symposium on Rock Mechanics, A. W. Khair, ed. Rotterdam: A. A. Balkema.
Yoshioka, N., and C. H. Scholz. 1989a. Elastic properties of contacting surfaces under normal and shear loads. 1. Theory. Journal of Geophysical Research, 94:17681–17690.
Yoshioka, N., and C. H. Scholz. 1989b. Elastic properties of contacting surfaces under normal and shear loads. 2. Comparison of theory with experiment. Journal of Geophysical Research, 94:17691–17700.