Hydromechanical Behavior of Fractured Rocks

Transcription

1 Chapter 7 Hydromechanical Behavior of Fractured Rocks Robert Zimmerman Ian Main 7.1 Introduction To a great extent, it is the nearly ubiquitous presence of fractures that makes the mechanical behavior of rock masses different from that of most engineering materials. These fractures also cause the behavior of rock masses to differ from that of small laboratory-size rock samples. Most laboratory tests on rock samples are conducted on specimens that are intact, and so, by definition, do not contain fractures. But almost all rock masses contain fractures on a scale larger than that of laboratory samples, with typical fracture spacings that range from tens of centimeters to tens of meters. These fractures have a controlling influence on the mechanical behavior of rock masses, since existing fractures provide planes of weakness on which further deformation can more readily occur. Fractures also often provide the major conduits through which fluids can flow. The field-scale permeability of a fractured rock mass may be many orders of magnitude larger than the permeability that would be measured on an intact core-scale specimen from the same field. The hydromechanical behavior of rock fractures can be studied on the scale of a single fracture and also on the scale of a fractured rock mass that contains many fractures. Obviously, the behavior of single fractures must be thoroughly understood before the behavior of fractured rock masses can be understood. The mechanical, hydraulic, and seismic behaviors of a single rock fracture are now fairly well understood. Each of these properties depends almost exclusively on the geometry of the fracture void space, which is discussed in Section The normal stiffness of a fracture is defined and discussed in Section 7.2.2, and the shear stiffness is treated in Section The hydraulic transmissivity of single 361

2 362 Chapter 7 Hydromechanical Behavior of Fractured Rocks rock fractures is examined in Section Coupling between the mechanical and hydraulic properties of a fracture are treated in Section Section 7.3, focuses on the genesis and growth of single fractures and fracture networks and on the macroscopic properties of fractured rock masses. The basic concepts of linear elastic fracture mechanics, which were introduced in Section 2.4, are extended in Section to apply to materials containing collections of cracks. Analytical and numerical models for the growth and coalescence of fractures are discussed in Section Scaling properties of fault and fracture systems are treated in Section Finally, the influence of fracture network geometry and topology on fluid flow and transport processes are briefly discussed in Section Hydromechanical Behavior of a Single Fracture Geometry of Rock Fractures An idealized rock fracture or joint consists of two nominally planar, rough surfaces. The surfaces are typically in contact with each other at some locations, but separated at others. The distance of separation, usually measured perpendicular to the nominal fracture plane, is known as the aperture. If the fracture has undergone substantial shear, it is usually classified as a fault; otherwise, it is denoted as a joint (Mandl, 2000). The space between the two rock surfaces may be clean or may contain (a) fault gouge that has been produced by the shearing of the two faces of rock, (b) clay minerals, or (c) mineral coatings that have been precipitated from flowing pore fluids. The genesis of faults and fractures in rocks is discussed at length by Mandl (2000). The following discussion will focus on the hydromechanical behavior of existing fractures, rather than the generation of new fractures or the growth of existing ones. We begin with a discussion of various mathematical concepts and definitions that are used to characterize fracture surfaces and apertures, focusing on clean fractures that contain no in fill. Consider a nominally planar fracture that lies in the x-y plane. Fracture surfaces are typically well correlated at very large wavelengths, so that, even if the fracture has waviness at large scales, a nominal fracture plane can usually be defined locally. Two parallel reference planes can be drawn, one inside the lower region of rock, the other inside the upper region (Figure 7.1). The distance between these two planes is denoted by d. The lower rock surface is then described by a surface height function z 1 (x, y), and the upper surface by the function z 2 (x, y). The aperture, defined as the distance between the opposing rock surfaces, measured perpendicular to the two reference planes, is then given by h(x, y) = d z 1 (x, y) z 2 (x, y). (7.1)

3 7.2 Hydromechanical Behavior of a Single Fracture 363 Reference Plane 2 d z 2 h Surface Profile 2 Surface Profile 1 z 1 Reference Plane 1 Figure 7.1 Two rough-fracture surface profiles, separated by an aperture h, along with the two reference planes, separated by a distance d. In principle, if the two surface profiles were known, the aperture would be known exactly, through equation(7.1). Moreover, all relevant hydro mechanical properties of the fracture, such as its hydraulic transmissivity, shear and normal stiffnesses, etc., could in principle be found from the geometry, by solving the relevant solid or fluid mechanical problem. But this detailed geometric information is usually not known, and moreover, solution of the problem of elastic (or plastic) deformation of the contacting surface, or the problem of fluid flow through the fractures void space, is currently not computationally feasible for realistic fracture profiles. Hence, current practice is to try to characterize the fracture in terms of a small number of statistical parameters, and to develop theories that relate the properties of the fracture to this set of parameters. In doing so, the fracture profiles and aperture fields are often treated as random variables, and the actual fracture is viewed as one stochastic realization of a random process that has certain statistical properties. If the statistical properties of each realization of a stochastic process are the same, the process is said to be ergodic (Lanaro, 2000). In this case, statistical parameters such as the mean and variance can be calculated from a single realization. The most basic statistical property of a random variable such as one of the surface profile functions z(x,y) is the mean, defined by 1 µ z = lim z(x,y)dxdy E{z}, (7.2) A A A where A is the nominal area of the fracture in the x-y plane. With reference to a particular transect of the fracture, say at a fixed value of y, we could define 1 µ z = lim L L L 0 z(x,y)dx. (7.3) If the statistical properties of a function are invariant with respect to translation of the origin, the process is called homogeneous or stationary (Adler and Thovert, 1999). In this case, the degree of correlation between the value of z at one location x,

4 364 Chapter 7 Hydromechanical Behavior of Fractured Rocks and at another location displaced from x by an amount ξ, can be quantified by the autocovariance function, cov z (ξ) = E {(z(x) µ z )(z(x + ξ) µ z )} = E {z(x)z(x + ξ)} µ 2 z. (7.4) Evaluation of the autocovariance function at a lag distance of ξ = 0 yields the variance, σ 2 z cov z(ξ = 0) = E{z 2 (x)} µ 2 z, (7.5) the square root of which is the standard deviation, σ z. With regard to a surface defined over a region of the two-dimensional x-y plane, rather than a linear transect, the autocovariance can be defined as a function of the vector ξ, as follows (Adler and Thovert, 1999): cov z (ξ) = E{(z(x) µ z )(z(x + ξ) µ z )}=E{z(x)z(x + ξ)} µ 2 z, (7.6) where x = (x, y) and ξ = (ξ, η). If the surface is isotropic, the autocovariance will depend on only the length of the lag vector, ξ = ( ξ 2 + η 2) 1/2. In this case, no generality is lost by putting η = 0. For simplicity of notation, isotropy will be assumed henceforth, in which case x and ξ can be treated as one-dimensional variables. Another measure of spatial correlation is the variogram function, γ z (ξ), defined by γ z (ξ) = E{[z(x + ξ) z(x)] 2 }. (7.7) Expansion of the term inside the brackets and comparison with equations (7.4) and (7.5) shows that cov z (ξ) = σz γ z(ξ), (7.8) where the term 1 2 γ z(ξ) is often referred to as the semivariogram. The average slope of a surface z between two locations x and x + ξ is given by [z(x + ξ) z(x)]/ξ. The variance of the average slope is, by equation (7.5), given by { [z(x + ξ) z(x)] σslope 2 2 } (ξ) = E µ 2slope = γ z(ξ) 2 ξ 2 ξ 2, (7.9) where the last step makes use of definition (7.7) and the fact that the mean value of the average slope must vanish, by appropriate choice of the reference plane. The variogram is therefore closely related to the variance of the mean value of the surface slope taken over the lag distance. From definition (7.7), the variogram and hence the semivariogram should vanish at ξ = 0, although in practice this is often obscured by an inability to make measurements at sufficiently small scales. At sufficiently large lag distances, a fracture surface will usually become uncorrelated, in which case its autocovariance

5 7.2 Hydromechanical Behavior of a Single Fracture 365 goes to zero, and the semivariogram approaches the variance. The power spectrum of z(x) can then be defined as the Fourier transform of its autocovariance function: G z (k) = 1 cov z (ξ)e ikξ dξ, (7.10) 2π where k = 2π/λ is the wavenumber and λ is the wavelength. Two common models for the autocovariance are the exponential and Gaussian models: cov z (ξ) = σ 2 z exp ( ξ /ξ 0), cov z (ξ) = σ 2 z exp [ (ξ/ξ 0 ) 2]. (7.11) For an exponential autocovariance, the surface is effectively uncorrelated at distances greater than about 4ξ 0, whereas for the Gaussian model the correlation is negligible for ξ>ξ 0. The correlation length, for which several different definitions can be given, is the distance beyond which the correlation between z(x) and z(x + ξ)is negligible. For exponential or Gaussian autocovariances, the parameter ξ 0 gives an indication of the correlation length. From equations (7.10) and (7.11), the power spectra of the exponential and Gaussian models are exponential: Gaussian: G z (k) = σ 2 z π 1/ξ 0 (1/ξ 0 ) 2 + k 2, (7.12) G z (k) = ξ 0σz exp( k2 ξ0 2 /4). (7.13) A profile z(x) is said to be self-affine if z(λx) = λ H z(x) for some constant H, which is known as the Hurst exponent. A profile is statistically self-affine if z(x) is statistically similar to λ H z(λx). A self-affine profile has a power spectrum of the form G z (k) = Ck α, (7.14) where α = 2H + 1 (Adler and Thovert, 1999, pp ). Such a power spectrum has been observed for profiles of fractures in crystalline and sedimentary rocks, bedding plane surfaces, and frictional wear surfaces (Brown and Scholz, 1985a; Power and Tullis, 1991). In practice, a power law can apply only between a lower limit of k min = 2π/L, where L is the length of the profile, and an upper limit of k max = 2π/l, where l is the distance along the x-axis between successive measurements (i.e., the sampling interval) Normal Stiffness of Rock Fractures If a rock core containing a through-going fracture that is aligned more or less perpendicular to the axis of the core is tested under uniaxial compression, the length

6 366 Chapter 7 Hydromechanical Behavior of Fractured Rocks change measured between the two end plates will consist of two contributions: the deformation of the intact rock, and an excess deformation, δ, that can be attributed to the fracture (Goodman, 1976). This excess deformation is called the joint closure, and is defined to be a nonnegative number that increases as the joint compresses. If the initial length of the specimen is L, and the normal stress is σ, the incremental change in the overall length of the core can be expressed as dl = dl r dδ = L E r dσ 1 κ n dσ, (7.15) where E r is theyoung modulus of the intact rock and κ n, with dimensions of Pa/m, is the normal stiffness of the fracture. An apparent Young modulus of the fractured rock, E fr, could be defined, but it would not be a meaningful property of the rock, as its value would depend on the length of the specimen, i.e., 1 1 dl E fr L dσ = (7.16) E r Lκ n Goodman (1976) made measurements of joint closure as a function of stress on artificially induced fractures by measuring the displacement across the total length of an intact sample, and then repeating the measurement across the core after it had been fractured. Joint closure measurements were made for mated joints, in which the two halves of the core were placed in the same relative position that they occupied before the core was fractured, and on nonmated joints, in which the two surfaces were rotated from their initial positions relative to one another. The unmated surfaces allowed much greater joint closure and had much lower joint stiffness (Figure 7.2). The joint closure is a highly nonlinear function of stress and levels off to some asymptotic value at high values of the confining stress. Goodman related the joint closure to the stress through the following empirical relation: ( ) δ t ] σ = σ 0 [1 +, for σ σ 0, (7.17) δ m δ where σ = σ 0 is some initial, low seating stress, t is a dimensionless empirical exponent, and δ m is the maximum possible joint closure, approached asymptotically as the stress increases. Bandis et al. (1983) made extensive measurements of joint closure on a variety of natural, unfilled joints in dolerite, limestone, siltstone, and sandstone and found that cycles of loading and unloading exhibited hysteresis and permanent set that diminished rapidly with successive cycles. Barton et al. (1985) later suggested that the hysteresis was a laboratory artifact and that in situ fractures probably behave in a manner similar to the third or fourth loading cycle. Bandis et al. (1983) fit the joint closure with functions of the form σ = κ 0 δ 1 (δ/δ m ) = κ 0δ m δ δ m δ, (7.18)

7 7.2 Hydromechanical Behavior of a Single Fracture 367 Normal Stress, σ (MPa) (a) Intact Rock Rock with Mated Joint Rock with Unmated Joint Axial Displacement, L (mm) Normal Stress, σ (MPa) (b) Mated Joint Unmated Joint Joint Closure, δ (mm) Figure 7.2 Measurements made by Goodman (1976, p. 172) of joint closure on a granodiorite specimen: (a) axial displacement of intact core, core with mated joint, and core with unmated joint; (b) joint closure, computed by subtracting displacement for intact specimen from displacement of jointed specimen. where κ 0 is an empirical parameter. The joint closure is related to the normal stress by ( ) σ δ = δ m. (7.19) σ + κ 0 δ m The normal stiffness of the fracture is given by κ n = dσ dδ = κ 0 (1 δ/δ m ) 2, (7.20) which shows that κ 0 is the normal stiffness at low confining stress. The function proposed by Goodman reduces to equation (7.18) when t = 1 and σ σ 0. Many aspects of the normal closure of an initially mated fracture can be qualitatively explained by the conceptual model developed by Myer (2000), in which a fracture is represented by a collection of collinear elliptical cracks (Figure 7.3). The cracks have length 2a, the spacing between the centers of adjacent cracks is 2λ, the fractional contact area is c = 1 (a/λ), and the cracks can have an arbitrary distribution of initial aspect ratios. From the elasticity solution of Sneddon and Lowengrub (1969), the incremental joint closure due to a small increase in normal stress is 4λ(1 ν)σ ( πa ) 4(1 ν)aσ [ π ] δ = ln sec = πg 2λ πg(1 c) ln sec 2 (1 c). (7.21) The normal compliance of the joint is given by 1 κ n = dδ dσ = 4(1 ν)a πg(1 c) ln sec [ π 2 (1 c) ]. (7.22) At low stresses, the fractional contact area is small and the compliance will be large. As the normal stress increases, those cracks with smaller aspect ratios close up.

8 368 Chapter 7 Hydromechanical Behavior of Fractured Rocks σ σ σ 2a L δ 2λ σ (a) σ (b) σ Figure 7.3 (a) Schematic model of a fracture as an array of two-dimensional cracks of length 2a and spacing 2λ. (b) Unit cell of fractured and intact rock, showing definition of δ (Myer, 2000). Although this disturbs the periodicity of the array, it can be modeled approximately by assuming that a (the half-length of the open cracks) remains the same but λ (the mean spacing between adjacent cracks) increases, leading to an increase in c and a consequent decrease in joint compliance. Expanding equation (7.22) for small values of 1 c shows that as c increases, δ πa(1 ν)(1 c) σ 2G 1 κ n πa(1 ν)(1 c). (7.23) 2G So, as the contact area increases, the compliance goes to zero, and the joint stiffness becomes infinite, in accordance with experimental observations. This model also indicates a size dependence, in that (other factors, such as c, being equal) smaller crack size a leads to stiffer fractures. Pyrak-Nolte et al. (1987) made casts of the void space of a natural granitic fracture using a low-viscosity alloy (Wood s metal) under various normal stresses, at 3, 33, and 85 MPa. Myer (2000) took transects of these casts, and found that as the normal stress increases, in addition to complete closure of some cracks, the rock faces occasionally come into contact at isolated points within existing cracks, creating two cracks with half-lengths less than a. Hence, as the normal stress increases, the contact area c increases and the mean crack length a decreases. According to equations (7.22) and (7.23), both the increase in c and the decrease in a lead to higher joint stiffness. For fracture surfaces that are unmated, perhaps as a result of previous shear displacement, Bandis et al. (1983) found that the normal stress could be fit with an equation of the form ln(σ/σ 0 ) = Jδ, (7.24)

9 7.2 Hydromechanical Behavior of a Single Fracture 369 where σ 0 is an initial, small stress level at which the joint closure is taken to be zero and J is a constant with dimensions of 1/L. The normal stiffness associated with this stress-closure relationship is κ n = dσ = Jσ, (7.25) dδ which increases linearly with stress. Another conceptual model for the normal stiffness of a rock fracture is to treat the fracture surface as a rough elastic surface, and use Hertzian contact theory (Timoshenko and Goodier, 1970, pp ) to analyze the deformation of the contacting asperities. Greenwood and Williamson (1966) considered a single, rough elastic surface whose asperities each have radius of curvature R, with a distribution of peak heights φ(z ), where the height Z of an asperity is measured relative to a reference plane that is parallel to the nominal fracture plane and can conveniently be located entirely within the rock (i.e., below the lowest troughs of the fracture surface). A value of Z is associated with each local peak, of which there are assumed to be η per unit area of fracture in the undeformed (zero stress) state. The height of the highest peak, measured from the reference plane, is initially equal to d 0 (Figure 7.4a). If such a surface is pressed against a smooth elastic surface of area A, the density of contacts is given by n = N/A = η φ(z )dz. (7.26) d 0 δ As the distribution function φ(z ) vanishes for Z >d 0, by construction, the contact density is zero when the joint closure δ is zero. In the hypothetical situation in which all asperities were pressed flat against the upper flat surface, δ would equal d 0, so the integral in equation (7.26) would approach unity, and the fraction of asperities in contact, n/η, would reach unity. The fractional contact area of asperities is given by c = A contact /A = πrη (Z d 0 + δ)φ(z )dz, (7.27) d 0 δ and the average normal stress acting over the surface is σ = 4 3 ηr1/2 E d 0 δ where the reduced elastic modulus E is defined by (Z d 0 + δ) 3/2 φ(z )dz, (7.28) 1 E = 1 ν2 1 E ν2 2 E 2, (7.29)

10 370 Chapter 7 Hydromechanical Behavior of Fractured Rocks (a) d 0 δ Flat Surface, No Stress Flat Surface, Stress Surface Profile 1 Reference Plane 1 (b) z 2 Reference Plane 2 Surface Profile 2 d 0 h Surface Profile 1 z 1 Reference Plane 1 (c) d 0 δ Reference Plane, No Stress Reference Plane, Stress Composite Surface Profile z = z 1 + z 2 Reference Plane Inside Rock Figure 7.4 (a) Single rough profile in contact with a smooth surface; (b) two rough surfaces; (c) composite profile (Cook, 1992) and subscripts 1 and 2 denote the properties of the rough and smooth surfaces, respectively. Swan (1983) measured the topography of 10 different surfaces of Offerdale slate and showed that the peak heights of asperities followed a Gaussian distribution. Greenwood and Williamson (1966) showed that the upper quartile of a Gaussian distribution could be approximated by an exponential distribution of the form φ(z ) = 1 s exp( Z /s), (7.30) where s is the mean, as well as the standard deviation, of the exponential distribution. Equations ( ) lead in this case to ( ) σ ln (πrs) 1/2 se η = δ d 0, (7.31) s which has the same form as the empirical relation found by Bandis et al. (1983) for fractures with unmated surfaces. Comparison of equations (7.24) and (7.31)

11 7.2 Hydromechanical Behavior of a Single Fracture 371 shows that the model of Swan and Greenwood and Williamson predicts J = 1/s σ 0 = (πrs) 1/2 (se η) exp( d 0 /s). (7.32) Comparison of equation (7.25) and (7.32) shows that the normal stiffness is equal to σ/s, and therefore increases with stress, and is inversely proportional to the roughness of the fracture. The parameters appearing in this expression for σ 0 would be difficult to estimate in practice, and indeed R would not typically be the same for all asperities, as is assumed in the model. However, Olsson and Brown (1993) noted that, for a wide range of fractures, σ 0 varies in the relatively narrow range of MPa. Brown and Scholz (1985b, 1986) extended this model to the closure of two rough surfaces in contact. The variable Z was redefined to represent the summed heights of the two opposing surfaces, each measured relative to the appropriate reference plane (Figure 7.4b,c), and the effective radius of curvature was taken as R = R 1 R 2 /(R 1 + R 2 ), where R 1 and R 2 are the radii of curvature of the pair of contacting asperities. Assuming that the radii of curvature of the asperities are uncorrelated with the heights and that nearby asperities do not elastically interact with each other, they found σ = 4 R 3 η 1/2 E ψ (Z d 0 + δ) 3/2 φ(z )dz, (7.33) d 0 δ where the brackets denote the mean values taken over all contacting asperities, and ψ is a tangential-stress correction factor whose mean value is very close to unity. If the shear stress within a particular asperity becomes sufficiently large, the asperity will yield and undergo irreversible plastic deformation. Two contacting spherical asperities will begin to yield when the displacement (at that particular contact point) reaches a critical value given by (Greenwood and Williamson, 1966; Brown and Scholz, 1986) ( ) H 2 δ p CR E, (7.34) where R is the effective radius of curvature, C is a dimensionless constant on the order of unity, and H is the indentation hardness of the rock mineral. For crystalline plasticity, H 3Y, where Y is the yield stress. As the fracture compresses, the highest asperities will be plastically flattened first. The fractional area of plastic contact is given by A plastic /A = πrη (Z d 0 + δ)φ(z )dz. (7.35) d 0 δ+δ p Greenwood and Williamson (1966) suggested that plastic deformation becomes nonnegligible when the ratio of plastic contact area to total contact area, i.e., the ratio of the integrals in equations (7.27) and (7.35), reaches about 2 10%.

12 372 Chapter 7 Hydromechanical Behavior of Fractured Rocks Behavior of Rock Fractures Under Shear If a fracture is located in a rock mass with a given ambient state of stress, the traction acting across the fracture plane can be resolved into a normal component and a shear component. The normal traction gives rise to a normal closure of the fracture, as described in Section The shear component of the traction causes the two rock faces to undergo a relative deformation parallel to the nominal fracture plane, referred to as a shear deformation. However, a tangential traction also typically causes the mean aperture to increase, in which case the fracture is said to dilate. Dilation arises because the asperities of one fracture surface must by necessity ride up to move past those of the other surface. Hence, shear deformation of a fracture is inherently a coupled process in which both normal and shear displacement occur. Displacement parallel to the nominal fracture plane is called the shear displacement, and is usually denoted by u (Figure 7.5a). The displacement in the direction perpendicular to the fracture plane is known in this context as dilation, and is denoted by ν. Shear displacement is reckoned positive if it is in the direction of the applied shear stress, whereas the dilation is positive if the two fracture surfaces move apart from each other. A typical but idealized curve for the shear displacement as a function of shear stress, as would be measured under conditions of constant normal stress, is shown in Figure 7.5b. v τ τ u τ p τ r (a) τ (b) 1 κ s u p u r u Figure 7.5 (a) Schematic diagram of a fracture sheared under constant normal stress (Goodman, 1989, p. 163). (b) Shear stress as a function of shear displacement (Goodman, 1976, p. 174). The shear stress first increases in a manner that is nearly proportional to the shear displacement. The slope of this line is the shear stiffness, κ s. During this phase of the deformation, the two fracture surfaces ride over each other s asperities, causing dilation of the fracture, but little degradation to the surfaces (Gentier et al., 2000). A peak shear stress τ p is eventually reached, corresponding to the point at which the asperities begin to shear off, causing irreversible damage to the fracture surfaces. This peak shear stress is also known as the shear strength of the fracture. The displacement at the peak shear stress is known as the peak displacement, u p.

13 7.2 Hydromechanical Behavior of a Single Fracture 373 If the fracture continues to be deformed under conditions of controlled shear displacement, the peak shear stress will be followed by an unstable softening regime, during which the shear stress decreases to a value known as the residual shear stress, τ r. During this phase the asperities continue to be crushed and sheared off, the fractional contact area between the two surfaces increases, and the dilation continues but at a decreased rate. The level of displacement at which the shear stress first reaches its residual value is known as the residual displacement, u r. The behavior of a fracture under shear depends very strongly on the normal stress acting across the fracture. A highly schematic view of the manner in which the relationship between τ and u varies with normal stress is shown in Figure 7.6a (Goodman, 1976, p. 177). In this model, the shear stiffness is independent of the normal stress, but both the peak shear stress and the residual shear strength increase with increasing normal stress. This is roughly consistent with the experimental measurements made by Olsson and Barton (2001) on a granite fracture taken from Äspö in Sweden (Figure 7.6b). τ 6 (a) 1 κ s Increasing σ u (b) Shear Stress, τ (MPa) σ = 4MPa σ = 2MPa Shear Displacement, u (mm) Figure 7.6 (a) Effect of normal stress σ on the relationship between shear stress and shear displacement (Goodman, 1976, p. 177). (b) Measurements made by Olsson and Barton (2001) on a granite fracture from Äspö in Sweden. The variation of peak shear stress as a function of normal stress is called the shear strength curve. Patton (1966) found the following bilinear function for τ p as a function of σ (Figure 7.7): for σ<σ T : τ p = σ tan(φ b + i), (7.36a) for σ>σ T : τ p = C J + σ tan φ r. (7.36b) At low normal stresses, shear deformation is assumed to take place predominantly by asperities sliding over each other. At higher normal stresses, the fracture possesses a cohesion C J that is due to the inherent shear strength of the asperities and has an effective angle of internal friction of φ r <φ b + i. Trigonometric considerations show that the parameters in equations (7.36a, 7.36b) are related by

14 374 Chapter 7 Hydromechanical Behavior of Fractured Rocks τ p 30 C j (a) φ b + i φ r σ (b) Peak Shear, τ p (MPa) Normal Stress, σ (MPa) Figure 7.7 (a) Bilinear model for the peak shear strength of a joint; parameters defined as in equations (7.36a, 7.36b). (b) Peak shear stresses measured on cement replicas of a fracture in Guéret granite (Gentier et al., 2000) in the direction labeled by them as 30, and fit to a curve of the form given by equations(7.36). tan(φ b + i) tan φ r = C J /σ T. Jaeger (1971) proposed the following continuous function: ( τ p = 1 e σ/σ ) C J + σ tan φ r, (7.37) which asymptotically approaches equation (7.36a) and (7.36b) for small and large normal stresses, respectively. The parameter σ is a transition stress that roughly demarcates the two regimes but which is not numerically identical to the parameter σ T. The peak shear stresses measured by Gentier et al. (2000) on cement replicas of a fracture in Guéret granite show qualitative agreement with this type of model (Figure 7.7b). A simple mechanical model of two flat surfaces that have an intrinsic (or basic ) friction angle of φ b and whose interface is inclined by an angle i from the nominal fracture plane, leads directly to equation(7.36a). Recognizing that this model is oversimplified but that the coefficient i in equation(7.36a) must depend on the roughness of the fracture, Barton (1973) correlated i to the joint roughness coefficient (JRC), an empirical measure of roughness whose value is estimated by comparing a fracture surface profile with standard profiles whose roughnesses are labeled from Examination of data from fractures in various sedimentary, igneous, and metamorphic rocks led Barton to the correlation i = JRClog 10 (JCS/σ), (7.38) where JCS is the joint compressive strength, which is equal to the unconfined compressive strength of the intact rock for unweathered fracture surfaces but which has a much lower value for weathered surfaces (Barton and Choubey, 1977). Grasselli (2001) attempted to correlate i to more objectively quantifiable measures of roughness that can be estimated using optical means.

15 7.2 Hydromechanical Behavior of a Single Fracture Hydraulic Transmissivity of Rock Fractures In many rock masses, field-scale fluid flow takes place predominantly through joints, faults, or fractures, rather than through the matrix rock itself. In some cases most of the flow may take place through a single such discontinuity, which for simplicity will be referred to as a fracture, whereas in other cases the flow occurs through an interconnected network of such fractures. Fracture-dominated flow is of importance in many areas of technological interest. Nearly half of all known hydrocarbon reserves are located in naturally fractured formations (Nelson, 1985), as are most geothermal reservoirs (Bodvarsson et al., 1986). Fracture flow is of importance in understanding and predicting the performance of underground radioactive waste repositories (Wu et al., 1999). Indeed, it has become increasingly clear during the past few decades that fracture-dominated flow is the rule, rather than the exception, in much of the subsurface. On the scale of a single fracture, fluid flow is governed by the Navier-Stokes equations, which in their most general form can be written as (Batchelor, 1967, pp ) u t + (u )u = F 1 ρ p + µ ρ 2 u, (7.39) where u = (u x,u y,u z ) is the velocity vector, F is the body-force vector per unit mass, ρ is the fluid density, µ is the fluid viscosity, and p is the pressure. The Navier-Stokes equations embody the principle of conservation of linear momentum, along with a linear constitutive relation that relates the stress tensor to the rate of deformation. The first term on the left of equation(7.39) represents the acceleration of a fluid particle, because, at a fixed point in space, the velocity may change with time. The second term, the advective acceleration, represents the acceleration that a particle may have, even in a steady-state flow field, by virtue of moving to a location at which there is a different velocity. The forcing terms on the right side represent the applied body force, the pressure gradient, and the viscous forces. Often, the only appreciable body force is gravity, in which case F = ge z, where e z is the unit vector in the upward vertical direction and g = 9.81m/s 2.If the density is uniform, gravity can be eliminated from the equations by defining a reduced pressure, ˆp = p + ρgz (Phillips, 1991, p. 26), in which case F 1 ρ p = ρge z 1 ρ p = 1 ρ ( p + ρge z) = 1 ρ (p + ρgz) 1 ρ ˆp. (7.40) Hence, the governing equations can be written without the gravity term if the pressure is replaced by the reduced pressure. For simplicity of notation, the reduced pressure will henceforth be denoted by p. In the steady state, the Navier-Stokes equations then reduce to µ 2 u ρ(u )u = p. (7.41)

16 376 Chapter 7 Hydromechanical Behavior of Fractured Rocks Equation(7.41) represents three equations for the four unknowns: the three velocity components and the pressure. An additional equation to close the system is provided by the principle of conservation of mass, which for an incompressible fluid is equivalent to conservation of volume and takes the form divu = u = u x x + u y y + u z z = 0. (7.42) The compressibility of water is roughly / Pa (Batchelor, 1967, p. 595), so a pressure change of 10 MPa would alter the density by only 0.5%; the assumption of incompressibility is therefore reasonable. The set of four coupled partial differential equations (7.41) and (7.42) must be augmented by the no slip boundary conditions, which state that at the interface between a solid and a fluid the velocity of the fluid must equal that of the solid. This implies that at the fracture walls not only must the normal component of the fluid velocity be zero but also that the tangential component must vanish. The simplest conceptual model of a fracture, for hydrological purposes, is that of two smooth, parallel walls separated by a uniform aperture, h. For this geometry, the Navier-Stokes equations can be solved exactly, to yield a velocity profile that is parabolic between the two walls. If the x-axis is aligned with the pressure gradient, the y-axis taken perpendicular to the pressure gradient within the plane of the fracture, and the z-axis taken normal to the fracture plane, with the fracture walls located at z =±h/2, the solution to equations (7.41) and (7.42) is (Zimmerman and Bodvarsson, 1996) u x = 1 p 2µ x [ (h/2) 2 z 2], u y = 0, u z = 0. (7.43) The total volumetric flux, with units of m 3 /s, is found by integrating the velocity: Q x = w +h/2 h/2 u x dz = w p 2µ x +h/2 h/2 [(h/2) 2 z 2] dz = wh3 p 12µ x, (7.44) where w is the depth of the fracture in the y direction, normal to the pressure gradient. The term T = wh 3 /12 is known as the fracture transmissivity. As the transmissivity is proportional to the cube of the aperture, this result is known as the cubic law. The result T = wh 3 /12, which is exact only for smooth-wall fractures of uniform aperture, must be modified to account for roughness and asperity contacts. To do this rigorously requires solution of the Navier-Stokes equations for more realistic geometries. However, in general, the presence of the advective acceleration term (u )u renders the Navier-Stokes nonlinear and consequently very difficult to solve. An exact solution is obtainable for flow between smooth parallel plates only because the nonlinear term vanishes identically in this case: the velocity is

17 7.2 Hydromechanical Behavior of a Single Fracture 377 in the x direction and is thus orthogonal to the velocity gradient, which is in the z direction. In principle, the Navier-Stokes equations could be solved numerically for realistic fracture geometries, but computational difficulties have as yet not allowed this to be achieved. Consequently, the Navier-Stokes equations are usually reduced to more tractable equations, such as the Stokes or Reynolds equations. The Stokes equations derive from the Navier-Stokes equations by neglect of the advective acceleration terms, which is justified if these terms are small compared with the viscous terms. A priori estimates of the magnitudes of the various terms in the steady-state Navier-Stokes equations for flow through a variableaperture fracture can be achieved as follows (Zimmerman and Bodvarsson, 1996). Let U be a characteristic velocity, such as the mean velocity in the direction of the macroscopic pressure gradient. As the in-plane velocity varies quasi-parabolically from zero at the upper and lower surfaces to some maximum value on the order of U in the interior, the magnitude of the viscous terms can be estimated as (Figure 7.8a) µ 2 u u x µ 2 z 2 µu h 2. (7.45) z x Q h(x) (a) (b) Figure 7.8 (a) Schematic of a rough-walled fracture. (b) Parabolic velocity profile assumed in the derivation of the Reynolds equation(7.55). The term h 2 appears because the velocity is differentiated twice with respect to the variable z. Since the advective acceleration, or inertia, terms contain first derivatives of velocity, their order of magnitude can be estimated as ρ(u )u ρu2, (7.46) where is some characteristic dimension in the direction of flow, such as the dominant wavelength of the aperture variation, or the mean distance between asperities. The condition for the inertia forces to be negligible compared with the viscous forces is ρu 2 µu h 2 or Re ρuh2 1, (7.47) µ where the reduced Reynolds number Re is the product of the traditional Reynolds number, ρuh/µ, and the geometric parameter h/.

18 378 Chapter 7 Hydromechanical Behavior of Fractured Rocks If condition (7.47) is satisfied, which necessarily will be the case at sufficiently low velocities, the Navier-Stokes equations reduce to the Stokes equations µ 2 u = p, (7.48) which can be written in component form as 2 u x x 2 2 u y x 2 2 u z x u x y u y y u z y u x z 2 = 1 p µ x, + 2 u y z 2 = 1 p µ x, + 2 u z z 2 = 1 p µ x. (7.49a) (7.49b) (7.49c) Again, these three equations must be supplemented by the conservation of mass equation(7.42). The Stokes equations are linear and consequently somewhat more tractable than the Navier-Stokes equations. More importantly, if the flow is governed by the Stokes equations, the resulting relation between the volumetric flux and pressure gradient will be linear, in analogy with Darcy s law. Only under such conditions will the flux be given by Q = (T /µ) p, where the transmissivity T is independent of the pressure gradient and where Q and p each lie within the fracture plane. Various approaches, including more precise order-of-magnitude estimates that account for typical values of the parameter h/ (Oron and Berkowitz, 1998; Zimmerman and Yeo, 2000), numerical simulations of flow through simulated fracture apertures (Skjetne et al., 1999), and perturbation solutions for flow between a smooth wall and a sinusoidal wall (Hasegawa and Izuchi, 1983), each show that the relationship between flux and pressure gradient is nearly linear for Reynolds numbers less than about 10. For higher Reynolds numbers, a nonlinear relationship of the form p = µ Q + β Q 2 (7.50) T is observed. At a given flow rate, an additional non-darcy pressure drop is added to the Darcian pressure drop. Reynolds numbers greater than 10 are difficult to avoid in laboratory experiments if the flow rates and pressure drops are to be large enough to measure with sufficient accuracy (Witherspoon et al., 1980; Yeo et al., 1998). However, in most subsurface flow situations, the nonlinear term is negligible. This nonlinearity is not necessarily due to turbulence, which occurs only at much higher Reynolds numbers. The nonlinearity observed at values of Re is due to merely the effects of curvature of the streamlines (Phillips, 1991, p. 28) and occurs in the laminar flow regime.

19 7.2 Hydromechanical Behavior of a Single Fracture 379 Brown et al. (1995) and Mourzenko et al. (1995) have solved the Stokes equations numerically for a few simulated fracture profiles, but use of the Stokes equation for studying fracture flow is not yet common and is by no means computationally straightforward. Typically, the Stokes equations are reduced further to the Reynolds lubrication equation, which requires that the variations in aperture occur gradually in the plane of the fracture. The magnitudes of the second derivatives in equation(7.49a) can be estimated as 2 u x x 2 2 u x y 2 = U 2 2 u x z 2 = U h 2, (7.51) and similarly for equation(7.49b). If (h/ ) 2 1, the derivatives within the plane will be negligible compared with the derivative with respect to z, and equations (7.49a and 7.49b) can be replaced by 2 u x z 2 = 1 p µ x 2 u y z 2 = 1 p µ y. (7.52) Integration of both of these equations with respect to z, bearing in mind the noslip boundary conditions at the top and bottom walls, z = h 1 and z = h 2, yields (Figure 7.8b) u x (x,y,z) = 1 p(x, y) (z h 1 )(z + h 2 ), 2µ x (7.53a) u y (x,y,z) = 1 p(x, y) (z h 1 )(z + h 2 ). 2µ y (7.53b) This is essentially the same parabolic velocity profile as occurs for flow between parallel plates, except that the velocity vector is now aligned with the local pressure gradient, which is not necessarily collinear with the global pressure gradient. Integration across the fracture aperture, using a temporary variable ξ = z + h 2 that represents the vertical distance from the bottom wall, yields hu x (x, y) = hu y (x, y) = h1 u x (x, y, z)dz = h3 (x, y) h 2 12µ h1 u y (x, y, z)dz = h3 (x, y) h 2 12µ p(x, y), (7.54a) x p(x, y), (7.54b) y where h = h 1 + h 2 is the total aperture and the overbar indicates averaging over the z direction. Equations (7.54a) represent an approximate solution to the Stokes equations, but contain an unknown pressure field. A governing equation for the pressure field is found by appealing to the conservation of mass equation(7.42), which however, applies to the local velocities, not the integrated values. But u = 0, so the

20 380 Chapter 7 Hydromechanical Behavior of Fractured Rocks integral of u with respect to z must also be zero. Interchanging the order of the divergence and integration operations, which is valid as long as the velocity satisfies the no-slip boundary condition, then shows that the divergence of the z-integrated velocity, u, must also vanish. Hence, (hu x ) x + (hu y) y = 0, so ( h 3 p ) + ( h 3 p ) = 0, (7.55) x x y y which is the Reynolds (1886) lubrication equation. An important assumption in the derivation of the Reynolds equation is that the velocity profile is parabolic at every location (x, y) in the fracture plane. Dijk et al. (1998) used nuclear magnetic resonance imaging to measure velocity profiles in a limestone fracture having a mean aperture of about 2 mm at Reynolds numbers on the order of unity. They found most of the profiles to be roughly parabolic, although often the profiles were slightly asymmetric and in some cases the velocity showed two local maxima instead of one. The aperture h that appears in the Reynolds equation is usually intended to be measured in the z direction, perpendicular to the nominal fracture plane. If the fracture deviates appreciably from planarity in some macroscopic sense, Ge (1997) suggested measuring h normal to the local centerline. Mourzenko et al. (1995) suggested that h(x, y) be defined as the diameter of the largest sphere that could be placed within the fracture, centered at location (x, y). Oron and Berkowitz (1998) suggested that h is best interpreted not necessarily as a local parameter but rather as the distance between two smooth fracture surfaces for which the smallest-scale roughness has been averaged away (Figure 7.9). Insofar as flow through a fracture is accurately modeled by the Reynolds equation, the problem of finding the transmissivity of a variable-aperture fracture reduces to the well-studied problem of finding the effective conductivity of (a) (b) (c) (d) Figure 7.9 Various definitions for the fracture aperture h for use in the Reynolds lubrication equation. The two rock surfaces are indicated by the thick, jagged lines. Aperture (a) is the distance between the two rock surfaces measured normal to the nominal macroscopic fracture plane, aperture (b) is the distance between the rock surfaces measured normal to the local fracture plane, aperture (c) is the distance between the two smoothed-out versions of the fracture surfaces, and aperture (d) is the diameter of the largest sphere that can fit between the rock surfaces.

21 7.2 Hydromechanical Behavior of a Single Fracture 381 a heterogeneous two-dimensional conductivity field, with h 3 playing the role of the conductivity. This problem is conveniently discussed in terms of the hydraulic aperture, h H, which is defined so that T = wh 3 H /12. Using variational principles, it can be shown that the hydraulic aperture is bounded by (Beran, 1968, p. 242) h 3 1 h 3 H h 3, (7.56) where x x m is the arithmetic mean value of the quantity x. The lower bound, the so-called harmonic mean, corresponds to the case in which the aperture varies only in the direction of flow, whereas the upper bound, the arithmetic mean, corresponds to aperture variation only in the direction transverse to the flow (Neuzil and Tracy, 1981; Silliman, 1989). These bounds are theoretically important, because they are among the few results pertaining to flow in rough-wall fractures that are rigorously known, but they are usually too far apart to be quantitatively useful. For instance, it is invariably the case that the hydraulic aperture is less than the mean aperture, i.e., h 3 H h 3. But h 3 < h 3 for any nonuniform distribution, so the bounds alone are not sufficiently powerful to show that h 3 H h 3. Elrod (1979) used Fourier transforms to solve the Reynolds equation for a fracture with an aperture having sinusoidal ripples in two mutually perpendicular directions, and showed that, for the isotropic case, ( ) h 3 H = h σh 2 2 h (7.57) Zimmerman et al. (1991) considered the case of small regions of unidirectional ripples, which were then randomly assembled, and found results that agreed with equation(7.57) up to second order, for both sinusoidal and sawtooth profiles. Furthermore, equation(7.57) is consistent with the results of Landau and Lifshitz (1960, pp. 45 6), who required only that the aperture field be continuous and differentiable. An alternative expression that agrees with equation(7.57) up to second order, but which does not yield unrealistic negative values for large values of the standard deviation, is (Renshaw, 1995) ( h 3 H = h σ ) h 2 3/2 h 2. (7.58) Dagan (1993) expressed the effective conductivity of a heterogeneous twodimensional medium in a form that, in the context of fracture flow, can be written as ( ) ) h 3 H = e3 ln h 1 + a 2 σy 2 + a 4σY h 3 G (1 + a 2 σy 2 + a 4σY , (7.59)

22 382 Chapter 7 Hydromechanical Behavior of Fractured Rocks where Y = ln h, σ Y is the standard deviation of ln h, and h G exp ( ln h ) is the geometric mean of the aperture distribution. Using a perturbation method and the assumption of a lognormal aperture distribution, Dagan showed that the coefficients a n vanish at least up to n = 6, implying that the geometric mean is a very good approximation for the hydraulic aperture in the lognormal case. Dagan s result agrees with equation(7.57) up to second order in σ Y (Zimmerman and Bodvarsson, 1996). The predictions of equation(7.58) compare reasonably well with several numerical simulations and with some laboratory data (Figure 7.10a). Patir and Cheng (1978) used finite differences to solve the Reynolds equation for flow between two surfaces, the half-apertures of which each obeyed a Gaussian height distribution with linearly decreasing autocorrelation functions. Brown (1987) performed a similar analysis for simulated fractures having fractal roughness profiles. These profiles had fractal dimensions between D = 2, which corresponds to a fracture having smooth walls, and D = 2.5, which was found by Brown and Scholz (1985a) to correspond to the maximum amount of roughness that occurs in real fractures. The transmissivities computed by Brown were found to essentially depend on h and σ h, with little sensitivity to D; the data in Figure 7.10a are for D = 2.5. Figure 7.10b compares the predictions of equation(7.58) with values measured by Hakami (1989) in the laboratory on five granite cores from Stripa, Sweden. These five fractures had mean apertures that ranged from µm and relative roughnesses σ/ h of ; one additional fracture, with a mean aperture of 83 µm, had no measurable transmissivity. (h H /h m ) 3 (a) Patir & Cheng (1978) Brown (1987) Renshaw (1995) Relative Smoothness, h m /σ h H 3 (predicted) [10 12 m 3 ] (b) h 3 H (measured) [10 12 m 3 ] Figure 7.10 (a) Normalized transmissivity of simulated fractures. (b) Transmissivities measured by Hakami (1989) on fractures in granite. Both are compared with predictions of Brown (1987), i.e., equation(7.57), shown as the solid lines. In passing from the Stokes equations to the Reynolds equation, the momentum equation in the z direction, (7.49c), is ignored. The z-component of the pressure gradient vanishes, in the mean, as does the z-component of the velocity, but this does not necessarily imply that all of these terms are small locally. The error incurred by replacing the Stokes equations by the Reynolds equation is in some sense related to the extent to which the terms in equation(7.49c) are indeed negli-

23 7.2 Hydromechanical Behavior of a Single Fracture 383 gible. Visual examination of fracture casts shows that the condition (h/ ) 2 1, which is needed for these terms to be negligible, is not always satisfied. The problem is not aperture variation per se, but rather the abruptness with which the aperture varies.yeo et al. (1998) measured aperture profiles and transmissivities of a fracture in a red Permian sandstone from the North Sea and solved the Reynolds equation for this fracture using finite elements. They found that the Reynolds equation over predicted the transmissivity by 40% 100%, depending on the level of shear displacement. Similar results have been found for artificial fractures (Nicholl et al., 1999), implying that the Reynolds equation may in some cases be an inadequate model for fracture flow. Fluid flow through fractures is also hindered by the presence of asperity regions at which the opposing fracture walls are in contact, and the local aperture is consequently zero. Models such as the geometric mean and the harmonic mean predict zero transmissivity if there is a finite probability of having h = 0. A lognormal aperture distribution, on the other hand, does not allow for any regions of zero aperture. These facts suggest using the methods described above for the regions of nonzero aperture and treating the contact regions by other methods (Walsh, 1981; Piggott and Elsworth, 1992). If an effective hydraulic aperture, call it h 0, can be found for the open regions of the fracture, the effect of asperity regions can be modeled by assuming that the fracture consists of regions of aperture h = h 0, and regions of aperture h = 0. If the flow rate is sufficiently low, i.e., Re 1, and the characteristic inplane dimension a of the asperity regions is much greater than the aperture, i.e., h 0 /a 1, then flow in the open regions is governed by equation(7.55), with h 0 = h = constant, yielding Laplace s equation, 2 p 2 p x p = 0. (7.60) y2 Boundary conditions must be prescribed along the contours Ɣ in the x-y plane that form the boundaries of the contact regions. As no fluid can enter these regions, the component of the velocity vector normal to these contours must vanish. But the velocity vector is parallel to the pressure gradient, as shown by equations (7.54a, 7.54b), so the boundary conditions for equation(7.60), along each contour Ɣ i, are p = ( p) n = 0, (7.61) n where n is the outward unit normal vector to Ɣ i and n is the coordinate in the direction of n. This mathematical model of flow between two smooth parallel plates, obstructed by cylindrical posts, is known as the Hele-Shaw model (Bear, 1998, pp ). Boundary condition (7.61) assures that no flow enters the asperity regions, but the no-flow condition also requires the tangential component of the velocity to vanish, i.e., ( p) t = 0, where t is a unit vector in the x-y plane perpendicular to n.

24 384 Chapter 7 Hydromechanical Behavior of Fractured Rocks However, it is not possible to impose boundary conditions on both the normal and tangential components of the derivative when solving Laplace s equation (Bers et al., 1964, pp ). So, solutions to the Hele-Shaw equations typically do not satisfy ( p) t = 0 and therefore do not account for viscous drag along the sides of the asperities. The relative error induced by this incorrect boundary condition is approximately h/a (Thompson, 1968; Kumar et al., 1991). Pyrak- Nolte et al. (1987) observed apertures in a fracture in crystalline rock that were m and asperity dimensions (in the fracture plane) that were m. Gale et al. (1990) measured apertures and asperity dimensions on a natural fracture in a granite from Stripa, Sweden, under a normal stress of 8 MPa and found h 0.1 mm and a 1.0 mm. These results imply that viscous drag along the sides of asperities will be negligible compared with the drag along the upper and lower fracture walls, consistent with the assumptions of the Hele-Shaw model. Walsh (1981) used the solution for potential flow around a single circular obstruction of radius a (Carslaw and Jaeger, 1959, p. 426) to develop the following estimate of the influence of contact area on fracture transmissivity: h 3 H = 1 c h c, (7.62) where c is the fraction of the fracture plane occupied by asperity regions. This expression has been validated numerically (Zimmerman et al., 1992) for asperity concentrations up to 0.25, which covers the range of contact areas that have been observed in real fractures (Witherspoon et al., 1980; Pyrak-Nolte et al., 1987). If the contact regions are randomly oriented ellipses of aspect ratio a 1, then (Zimmerman et al., 1992) h 3 H = 1 βc (1 + h3 α)2 0, where β =. (7.63) 1 + βc 4α As the ellipses become more elongated, the factor β increases and the hydraulic aperture decreases. Although contact areas are not perfectly elliptical, equation(7.63) can be used if the actual asperity shapes are replaced by equivalent ellipses having the same perimeter and area ratios Coupled Hydromechanical Behavior As both the mechanical and hydraulic behaviors of rock fractures are controlled to a great extent by the morphology of the fracture surfaces, it is to be expected that the stiffnesses (normal and shear) and transmissivity of a fracture should be related to one another in some way (Cook, 1992; Pyrak-Nolte and Morris, 2000). Although this is undoubtedly true, the relationship is indirect and very complex, and no simple correlations seem to exist between the mechanical and hydraulic properties. This is because hydraulic transmissivity depends primarily

25 7.2 Hydromechanical Behavior of a Single Fracture 385 on the aperture of the open areas of the fracture and to a lesser extent on the contact area, whereas normal stiffness depends mainly on the amount and distribution of the contact areas (Hopkins, 2000). As the normal stress on a fracture increases, the mean aperture decreases, causing the transmissivity to decrease. Because of roughness and asperity contact, however, the change in mean aperture is not exactly equivalent to the joint closure defined in Section Furthermore, although transmissivity is proportional to mean aperture cubed, it also depends on the variance of the aperture and the amount of contact area, through relations such as (7.58) and (7.63). Consequently, the transmissivity of a fracture that is deforming under a normal load is not always directly proportional to the cube of the mean aperture. Witherspoon et al. (1980) measured the transmissivity of a tensile fracture in marble while compressing it under a normal load (Figure 7.11). At low stresses, the aperture is large and the relative roughness σ/ h is low. The fracture therefore approximates the parallel plate model, and the transmissivity varies with the cube of the mean aperture (region I). As the stress increases, in the open areas of the fracture the mean aperture will decrease while σ remains nearly the same (Renshaw, 1995), causing the transmissivity to decrease faster than the cube of the mean aperture, as shown by, say, equation(7.58). In other areas, regions of the fracture will come into contact. This phenomenon also causes an additional decrease in transmissivity; see equations (7.62) and (7.63). The combination of a decrease in mean aperture, increase in relative roughness, and increase in contact area leads to the transmissivity increasing faster than the cube of the mean aperture (region II). In this region the relation can be approximated by a power law with an exponent greater than 3 (Pyrak-Nolte et al., 1987), often as high as However, this power law holds over a very narrow range of mean apertures, typically only a factor of 2 or Mean Aperture ( m) III II I Transmissivity (m 3 ) Increasing Stress Decreasing Stress Cubic Law Figure 7.11 Transmissivity of a marble fracture as a function of mean aperture, as measured by Witherspoon et al. (1980). Data points are from three different loading cycles, although for clarity not all points are shown.

26 386 Chapter 7 Hydromechanical Behavior of Fractured Rocks Sisavath et al. (2003) modeled the transition between regions I and II by considering a fracture with a sinusoidal aperture variation. As the normal stress increases, the two surfaces are assumed to move toward each other, giving a decrease in h while the roughness, σ, remains constant. Flow was modeled by a perturbation solution of the Stokes equations, including terms up to (σ/λ) 2, where λ is the wavelength. Plausible values of initial roughness and wavelength, such as σ/ h = 0.5 and λ/ h 5, lead to slopes of 3 4 at high mean apertures but which abruptly increase to 8 10 as mean aperture decreases. As normal stress increases further, the mean aperture continues to decrease, but the transmissivity stabilizes at some small but nonzero residual value (region III). Cook (1992) suggested the following qualitative explanation for the existence of a residual transmissivity. Metal castings of the void space of a natural granite fracture under a range of stresses (Pyrak-Nolte et al., 1987) revealed, at high stresses (85 MPa), the existence of large oceanic regions of open fracture, connected by tortuous paths through archipelagic regions filled with numerous small, closely spaced contact regions. As the stress increases, the oceanic regions, necessarily having very low aspect ratios, will continue to deform, leading to a continued decrease in mean aperture. The resistance to flow, however, is controlled by the small tortuous channels that connect the oceanic regions. These channels necessarily have relatively large aspect ratios and are therefore extremely stiff. For example, the channels observed by Pyrak-Nolte et al. (1987) had widths of less than 100 µm (Myer, 2000). The mean aperture at high stresses was approximately 10 µm, so the aspect ratios of these channels was approximately 0.1. If the Young modulus of the intact rock is in the range of GPa, the apertures of such channels will decrease less than 1% under an additional stress of a few tens of megapascals. Hence, the result is a residual fracture transmissivity that is nearly stress independent. As a fracture undergoes shear displacement, on the other hand, the fracture will dilate and the transmissivity will increase. Olsson and Brown (1993) measured the hydraulic transmissivity of a fracture in Austin chalk while either increasing and decreasing the normal stress, keeping the shear displacement constant, or increasing the shear displacement, holding the normal stress constant. The flow geometry was annular radial, with fluid entering the fracture plane at the inner radius of 24.0 mm and leaving at the outer radius of 60.3 mm. At a fixed value of shear displacement, the transmissivity decreased with increasing normal stress, and then increased as the normal stress was removed, although some hysteresis is observed (Figure 7.12a). When the fracture was subjected to 3.5 mm of shear displacement at constant normal stress (equal to 4.3 MPa in Figure 7.12b), the joint dilated (i.e., negative joint closure) at a rate of about 50 µm per millimeter of shear displacement and the transmissivity increased by about 2 orders of magnitude. Qualitatively similar results were found by Yeo et al. (1998) for a red Permian sandstone fracture from the North Sea, and by Chen et al. (2000) for a granitic fracture from Olympic Dam mine in Central Australia.

27 7.2 Hydromechanical Behavior of a Single Fracture 387 Transmissivity (m 3 ) Offset = 3 mm Offset = 1 mm Offset = 0 mm (a) Normal Stress (MPa) T (10 12 m 3 ) τ (MPa) δ (10 4 m) (b) Shear Displacement (mm) Figure 7.12 Transmissivity of a fracture in Austin chalk, as measured by Olsson and Brown (1993): (a) Transmissivity as a function of normal stress, for different values of shear offset, with the open circles denoting increasing stress and the closed circles decreasing stress; (b) shear stress τ, transmissivity T, and joint closure δ, as functions of shear displacement, with normal stress held at 4.3 MPa. Esaki et al. (1999) subjected an artificially split fracture in a granite from Nangen, Korea, to shear displacements up to 20 mm, under various values (1, 5, 10, and 20 MPa) of normal stress. Flow was measured from a central borehole to the outer boundaries of a rectangular fracture plane of dimensions 100 mm 120 mm. Transmissivity typically decreased very slightly for the first mm or so of shear, until the peak shear stress was reached. It then increased by about 2 orders of magnitude as the shear offset increased to about 5 mm. After this point, when the shear stress had reached its residual value, the transmissivity essentially leveled off (Figure 7.13). When the shear displacement was reversed, the transmissivity decreased, but not to its original value, leaving an excess residual transmissivity at zero shear displacement. This hysteresis was larger at larger values of the normal stress. Transmissivity (m 3 ) Increasing Shear Decreasing Shear Normal Stress = 1 MPa (a) Shear Displacement (mm) Transmissivity (m 3 ) Increasing Shear Decreasing Shear Normal Stress = 10 MPa (b) Shear Displacement (mm) Figure 7.13 Transmissivity of a granitic joint as a function of shear displacement, as measured by Esaki et al. (1999) under a normal stress of (a) 1 MPa, and (b) 10 MPa.

28 388 Chapter 7 Hydromechanical Behavior of Fractured Rocks 7.3 Mechanical and Structural Properties of Fracture Populations The last section described the properties of individual fractures. In nature, fractures typically grow in herds, i.e., as individuals in a larger population. Fractures have to start somewhere, nucleating at local heterogeneities or stress concentrations as described in Chapter 1, Section 1.4. There are many such nucleation points in a composite ceramic material such as a rock, so there will generally be several nucleation points. Each will then grow as an isolated fracture at a rate that depends on the local stress, the fracture length itself, the local material strength, and the rate of chemical weakening reactions at the crack tip. Once the fracture grows past a certain length it will come under the influence of other local fractures, leading to coalescence into larger structures, where the interaction may lead to a further stress concentration. At the same time, other fractures may become inactive when they fall in the stress shadow of a neighboring crack. The geometrical arrangement of fractures, such as their size-frequency distribution, their spatial correlation, and the degree to which they are connected to form a coherent pathway or barrier to fluid flow, all depend explicitly on this evolutionary mechanical process. It is therefore important to outline the general principles of this evolution, before going on to describe the structural properties of natural fracture patterns and their influence on fluid flow. The material in this section, for completeness, reprises some results that are described in more detail in Chapter Basic Concepts in Fracture Evolution In this section we consider the problem of fracture growth using the quasi-static approximation, i.e., neglecting inertia. For slow crack growth by chemically assisted processes such as stress corrosion, this is a reasonable assumption. However, even if individual ruptures are dynamic, their duration is short compared with the overall evolutionary time scale of fracture growth, and hence the dynamic phase can be included, with some caveats, in a time-integrated quasi-static model. This approximation is reasonable to first order, if the main aim is to describe the structural properties of a fracture population. Nucleation Griffith (1920) was the first to realize that all fractures grow from preexisting weaknesses in composite materials. Using glass rods, stretched by a dead weight at their base, he cut controlled flaws into the glass in the form of notches of different lengths. Samples with larger notches broke at lower values of dead weight. This demonstrates that the stress required to nucleate a fracture (here, the stress is the weight of the dead load divided by the cross-sectional area of the rod) depends on its preexisting length and is not in itself a material property. Formally, the Griffith

29 7.3 Mechanical and Structural Properties of Fracture Populations 389 nucleation criterion can be derived thermodynamically by considering a crack of half-length c embedded in a perfect elastic material of Young modulus E and specific surface energy γ. The specific energy is the energy required to create a unit area of free surface of the material of interest. For this problem the Gibbs free energy F is the sum of the energy supply, U, held in the volume of the intact elastic material and in the experimental loading configuration, and the energy demand, Ɣ, required to create the fracture surface on an area A, i.e., F = U + Ɣ. For a small increment of crack growth da, the equilibrium criterion is F A = U A + Ɣ = 0. (7.64) A For a two-dimensional elliptical crack in a block of thickness t, where t c, under a pure tensile stress σ r at the remote boundary, the free energy is F = 4γct πσ2 r c2 t E. (7.65) Figure 7.14 illustrates this function, which has only one solution to equation(7.64) where F is a maximum. Thus, the equilibrium position, where F/ c = 0, is unstable. If A = 2ct, this reduces to the nucleation criterion G = U A = πσ2 r c = 2γ, (7.66) E where G is the energy release rate per unit surface area. This nucleation criterion also defines an instability criterion, since the crack can grow without limit once the stress or the crack length are high enough. If γ is the fundamental material property, 0.5 F F 0.4 F c c/c0 (a) c/c0 (b) Figure 7.14 Schematic diagrams showing (a) the free energy as a function of normalized crack length for an ideal elliptical Griffith crack in an infinite medium. The nucleation criterion for an unstable equilibrium position is c = c 0. (b) The free energy for an ideal Griffith crack in a medium with ideal periodic heterogeneity. In the latter case subcritical crack growth can occur in discrete steps c by thermal fluctuations that overcome the local free energy barrier F between stable states.

30 390 Chapter 7 Hydromechanical Behavior of Fractured Rocks then the strength, σ c, defined as the critical stress at failure, scales inversely as the square root of the preexisting crack length. This dependence was confirmed by Griffith s experiments and a variety of observations since. Growth Ideal Griffith cracks, once nucleated, would never stop growing. However, in any natural composite material the specific surface energy is not completely uniform. The heterogeneity can be expressed as a specific surface energy that depends on position: γ = f(x), where x is the position vector. Lawn (1993) summarizes a simple model for quasi-static fracture growth in such a material. In this latticetrapping model, the heterogeneity is due to fluctuations in the potential energy surface between atoms in a crystalline lattice and is hence periodic. However, similar heterogeneity on a larger scale could equally well apply at the typical grain size or bedding spacing, for example. In fact, the low specific surface energies for rocks (approximately 1 Jm 2 ) are several orders of magnitude lower than those required to form a free surface on a crystalline lattice (Scholz, 1990, p. 3). This implies that the controlling processes for fracture growth in rocks (as opposed to individual crystals) depend on preexisting flaws within and between individual grains in the polycrystalline lattice. To represent such heterogeneity in the simplest possible way, a periodic fluctuation at a spatial frequency k can be used: F(x) = F 0 + F 1 (cos kx), (7.67) Where F 0 is given by equation(7.65) and F 1 is the amplitude of the spatial fluctuation. In the general case the material heterogeneity could be expressed as a Fourier sum: F(x) = F 0 + F j e ik j x. (7.68) The introduction of heterogeneity such as equation(7.67) leads to the existence of many local extrema in the free energy (Figure 7.14b). Thus, we have many unstable nucleation points at maxima, and many stable points at minima, not just a single maximum. The crack can grow by an increment of length c, from one local minimum to the next stable minimum, by thermally activated chemical processes that overcome the local free energy barrier F. For a first-order chemical reaction, reaction rate theory predicts that the time this takes depends exponentially on the height of the barrier: τ = τ 0 e F/kT, (7.69) where τ 0 is a characteristic term (preexponential factor), k is the Boltzmann constant, and T is absolute temperature. For a periodic heterogeneity, the height of the barrier decreases approximately linearly with crack semilength ( F α βc) j=1

31 7.3 Mechanical and Structural Properties of Fracture Populations 391 until it vanishes at the inflection point representing the unstable nucleation of dynamic failure in the material (Lawn, 1993). With this approximation, τ = τ 0 e α β c = τ e β c, (7.70) where α = α/kt, β = β/kt, and τ = τ 0 e α. For periodic heterogeneity, the distance between stable minima c is approximately constant, so the velocity of crack growth in the quasi-static regime takes the following exponential form: V(c)= dc dt c τ(c) = V 0e β c, (7.71) where V 0 = c/τ. At constant stress, G is proportional to c, whence V (G) = V 0 e G/G 0. (7.72) In fact, experimental data on quasi-static, chemically assisted crack growth is often fitted empirically to the power law form known as Charles law (Atkinson, 1987): V (G) = V 0 (G/G 0 ) n, (7.73) where n is an exponent describing the nonlinearity of fracture growth. For a constant stress, this differential equation can be solved for c(t) to give, for the specific case n>1, c(t) = c 0 (1 t/t f ) ν, (7.74) where ν = 1/(n 1), and t f is a nominal failure time, defined as the time at which c becomes singular. This approximation holds as long as the local slope dc/dt is small compared with the velocity of sound in the medium. Failure would occur before t f if inertial terms were included or the finite sample size was taken into account. What might be the origin of the power law form? For the case of a single crack, it may be result from a specific form of nonperiodic material heterogeneity. For a fracture population, we now show that this arises more simply and spontaneously from the coalescence of fractures in a heterogeneous medium. Coalescence We illustrate the properties of fracture coalescence using a simple percolation model. Before doing this we reprise some of the critical properties of percolation (Stauffer and Aharony, 1994). Imagine a grid of elements with different random strengths, corresponding to local fluctuations in strength within each cell. As the stress is increased, an element is deemed to fail when the local stress exceeds the strength. No mechanical interactions are considered in a percolation model. The probability p of an element having failed is equal to the number of failed

32 392 Chapter 7 Hydromechanical Behavior of Fractured Rocks elements divided by the total number of elements. Initially the weakest elements fail as isolated sites distributed throughout the medium; this corresponds to the nucleation phase. As p increases, it becomes more difficult for new failed sites to avoid the boundaries of already failed elements, leading to the development of clusters of failed elements; this phase corresponds to the growth phase. Ultimately, the clusters themselves become so large that they also cannot avoid each other, leading to a phase of rapid crack growth by the mechanism of cluster coalescence. The largest cluster at any time defines the correlation length ξ, defined by ξ 2 = r ( ) r 2 g(r) / r g(r), (7.75) where r is the distance between two sites and g is the correlation function, i.e., the probability that a site at distance r from one failed site has also failed and belongs to the same cluster. The correlation length thus roughly corresponds to the size of the largest cluster (Bruce and Wallace, 1989). In the percolation model the correlation length increases very nonlinearly with the probability of local failure, according to ξ(p) (p p c ) ν, (7.76) where p c is the critical probability for a cluster that spans the cell grid from one side to the other. For the two-dimensional site-percolation model, p c = 0.5 and ν = 4/3. The critical probability defines the percolation threshold. If the strength distribution is random, then the probability density p of an element having a given strength between zero and the maximum strength is a constant. Thus, p(σ) = σ 0 p (σċ)dσċ = const σ. (7.77) If the stress is increased linearly with time, then σ = σt, so equation(7.76) reduces to ξ(t) = ξ 0 (1 t/t c ) ν, (7.78) where t c = t(p c ) is a nominal failure time. So, the biggest crack grows in exactly the same form as predicted by equation(7.74), again by combining time dependence and material heterogeneity. However, the material heterogeneity here is uniformly random it does not require preexisting correlations in the form of specific distributions of F(x), as would be the case for the growth of a single isolated crack. The percolation model (Stauffer and Aharony, 1994) also predicts a cluster size-frequency distribution of the form ( n(s) = S τ exp S/aξ D), (7.79)

33 7.3 Mechanical and Structural Properties of Fracture Populations 393 where n is the number of clusters of area S, a is a proportionality constant, and D is the fractal dimension of the clusters (as defined by Mandelbrot, 1982), which takes account of the roughness of the cluster perimeter. For Euclidean rupture areas, such as ellipses or circles, a = 1 and D = 2. For a two-dimensional percolation model, D = 1.89; in three dimensions, D = 2.5 (Stauffer and Aharony, 1994). Thus, the correlation length also influences the size-frequency distribution. For sizes smaller than the correlation length, the distribution is a power law. If the correlation length is small, then the power law fraction is negligible and the distribution approximates a single exponential. The combination of power law and exponential behavior implies that the first moment of the distribution is proportional to the standard gamma distribution in statistics. This behavior holds as long as the correlation length is small compared with the overall dimension L of the grid of elements, i.e., below the percolation threshold. Above the percolation threshold, the frequency for large clusters can exceed that predicted from the linear extrapolation of frequencies for the smaller ones. Such finite-size effects also influence our interpretation of natural fracture patterns, as we shall see in the following Mean-Field (Analytical) Models for Fracture Growth The percolation model previously described for a fracture population predicts specific forms for the scaling distributions of size, frequency, and time in response to the application of an external stress. However, the physical interactions between cracks are completely neglected. In this section we examine the effect of crack crack interactions on the growth of a fracture population. These models lump the individual contributions of individual cracks together, rather than calculating each interaction and summing them and hence are referred to here as mean-field models. Again, there are two traditions, one following from the single-crack tradition and one from the tradition of considering a fracture population. The Plastic Process Zone The main problem with the Griffith theory is that it predicts a stress singularity at the crack tip. Calculations from fracture mechanics show that the local stress tensor σ ij decays away from the crack tip in the form σ ij = f ij (θ)σ r (c/r) 1/2, (7.80) where r is the position and θ is the angle (relative to the crack axis) from the crack tip and f is a well-defined function of θ (Atkinson, 1987; Pollard and Segall, 1987). The stress intensity K = Yσ r c 1/2 where Y is a dimensionless geometrical factor that depends on the loading configuration, measures the strength of the singularity, or the size of the stress concentration at the crack tip. When compared with equation(7.66), it can be seen that K G, so the fracture mechanics formulation

34 394 Chapter 7 Hydromechanical Behavior of Fractured Rocks is directly consistent with the Griffith nucleation theory. In the derivation of equation(7.65) Griffith used the following result for the crack displacement profile: u(x) = 2σ r c E 2 x 2, (7.81) where the crack center is defined at x = 0 and the constant 2 is appropriate for plane stress conditions. Thus, an ideal Griffith crack has an elliptical profile, with a maximum displacement given by u max = 2σ rc E. (7.82) equation(7.82) applies to a stable Griffith crack in a uniform medium and predicts a critical energy release rate G c = 2γ from equation(7.66). If the specific surface energy γ is a constant material property, then the combination of equations (7.66) and (7.82) implies that ( γ u max = 2 πe ) (2c) 1/2. (7.83) Thus the maximum displacement scales as the square root of the fracture length L = 2c (Scholz, 1990). However, as soon as a finite heterogeneity is present, there are many potential nucleation points, all with different local values of the energy release rate, denoted G l c (Figure 7.14b), where Gl c is proportional to the local value of c, whence u max L. This implies scale-invariant crack growth, as observed in natural systems (Scholz, 1990). In summary, ideal Griffith cracks in a uniform medium would have displacement proportional to the square root of crack length, and those in a heterogeneous medium would have scale-invariant growth, with displacement proportional to length. This is significant since, if the hydraulic cubic law discussed in Section 7.2 holds, fluid flow is then dominated by the largest cracks, since they have both the largest aperture and greatest potential for connecting the inflow and outflow boundaries of the sample. In reality, stresses cannot be infinite at the crack tip r = 0. Dugdale (1960) instead postulated the existence of a process zone ahead of the crack tip, where the large stresses concentrated near the crack tip could be dissipated by thermally activated plastic yield in sheet metal samples. In his model, the microscopic and macroscopic rheology in the process zone is plastic. For silicates in the upper 10 km or so of the Earth s crust, the activation energies for local plastic processes are too high, so the local model of failure is brittle. However, for a cloud of microcracks growing in a distributed cloud ahead of the crack tip, the ensemble effect can be considered plastic. This corresponds to a mean-field model where the effect of the multiple crack crack interactions in the process zone is replaced by a single function, the yield stress. This illustrates a general point in rock mechanics, namely, that it is not necessary to have a one-to-one correspondence between microscopic and macroscopic rheology. What matters is whether failure is brittle (localized, involving a stress drop) or plastic (distributed, with yield occurring at constant stress) at the scale of interest.

35 7.3 Mechanical and Structural Properties of Fracture Populations σ u/umax 1 1 σy r*/c r/r* 3 (a) 0 0 x/c 1 (b) Figure 7.15 (a) Plot of local stress σ at position r from the crack tip for the Griffith crack (solid line) and a model with a process zone (dashed line). For the Griffith crack, the maximum stress is infinite at the crack tip. For the process zone model the maximum stress is the yield stress σ y at a characteristic distance r = r. (b) Schematic plot of the normalized opening displacement as a function of position x from the crack center of the two models. The difference between the Griffith and the Dugdale models is illustrated in Figure 7.15, after Cowie and Scholz (1992). Note that the stress singularity disappears in the Dugdale model and the shape of the displacement profile has a bell-shaped rather than an elliptical taper near the crack tip. A simple model for the process zone can be written in the form σ ij = f ij (θ)σ r (c/r) 1/2 r r, σ ij = σ y r r, (7.84) where r is the size of the process zone. From these two equations, at r = r : σ ij (r ) = f ij (θ)σ r (c/r ) 1/2 = σ y. (7.85) If the remote stress and the yield stress are each constant, then it follows from equation(7.85) that, for a given angle θ, c/r is also constant. That is, the size of the inelastic process zone scales linearly with its length. With this condition, Cowie and Scholz (1992), in a full treatment based on the Dugdale model, show that u max L σ y E, (7.86) where L = 2c + 2r is the total crack length, including the tapered part in the process zone at either crack tip. Thus, the displacement scales linearly with crack length, similar to the solution for a Griffith crack in a heterogeneous medium previously calculated. equation(7.86) holds when r is small compared with c.

36 396 Chapter 7 Hydromechanical Behavior of Fractured Rocks Damage Mechanics The single-crack model provides a good physical model for the scaling of displacement with crack length but requires damage to occur in only a small zone ahead of the crack tip. Such conditions are applicable to the growth of single tensile cracks with a precut starting notch and guiding groove in the double-torsion experiments of Atkinson and Meredith (1987) or for highly localized failure in the Earth. What then if we have multiple nucleation sites in the more general case of a heterogeneous sample? In this case the zone of damage is distributed throughout the sample volume, and the concept of a localized process zone becomes inappropriate. Instead, mechanical and material properties are calculated by considering a damage parameter that depends on the mean crack length c. For the example of tensile loading, Nπ c 2 =, (7.87) A s where A s is the cross-sectional area of the sample and N is the number of cracks. In an undamaged material = 0, and in a completely failed material = 1. It is important to emphasize that any mean-field theory does not imply that physical interactions are neglected, as say, for the percolation model previously described. Instead, physical interactions are averaged and replaced by a single modifier term that is independent of position, in contrast to, say, equation(7.80). In this limit, the effective stress on the undamaged material can be approximated by σ u = σ 1. (7.88) The stress strain relation is then σ u = Eε, where E is the Young modulus of the undamaged material, so that σ = E 0 (1 )ε = E ε, (7.89) where E is the effectiveyoung modulus of the damaged material. The rate of damage accumulation can be calculated from an ensemble form of equation (7.74) as c (t) = c 0 (1 t/t f ) ν : 0 < <1. (7.90) Figure 7.16 shows the behavior of equations ( ). The effective failure time t f defined for = 1 is again slightly less than t f. A more complete treatment of the principles and the variety of time-dependent damage models is reviewed by Costin (1987) Numerical Models for Fracture Growth In the foregoing discussion we have seen that simple physical, chemical, and statistical models predict exact physical forms for the acceleration of fracture length

37 7.3 Mechanical and Structural Properties of Fracture Populations σ/σmax Slope E/E0 0 0 ε 0.01 Figure 7.16 Stress strain relation σ(ε) for a damage mechanics model with damage accumulating according to an equation whose plot is of the form of Figure 7.15b. In this model the remote strain is increasing linearly with time and the elastic modulus E is decreasing from its initial value E 0, because of the damage being sustained by the material. A typical dimensionless breaking strain for rocks is approximate to failure, the size distribution of a fracture ensemble, the scaling of displacement to length, and the rheology of a multiply fractured material. None of these models, though, capture the aspect of fracturing most noticeable to the eye: that is, the spatial organization and pattern seen in field exposures. At this point we abandon the simplicity and tractability of analytical models and consider how elastic interactions that depend on position r from the crack tip affect the results in numerical models. Such models generally fall into two classes, equating to the site percolation and bond percolation models of Stauffer and Aharony (1994). The former is used to model the growth of fractures in the plane of fracture and the latter to fractures cutting across this plane. In-plane Models A simple form of the in-plane model is the cellular automaton. In this model the rock is modeled as a heterogeneous grid of elements, or sites, and the dependence on position is simplified to the extreme point of considering only nearest neighbors. Figure 7.17 shows an example of the development of damage by the two-stage process of (a) microcrack nucleation and growth and (b) microcrack coalescence, after Henderson et al. (1994.). The model preserves the negative feedback of localized tensile crack growth in a compressive stress field that is also a feature of Costin s (1987) mean-field model and also the positive feedback of stress concentration once cracks reach a certain size. In the model of Henderson et al. (1994.) crack growth is initially stabilized and then accelerated by a combined hardening and softening rule that depends on the number of elements in the cluster. This results in fractures initially avoiding each other and then coalescing once they reach a critical size. The local strength at time zero is distributed here according to a power law, leading to a spatial correlation in the strength field at time zero. The mechanical

38 398 Chapter 7 Hydromechanical Behavior of Fractured Rocks Zone of Damage Intact Rock Fracture Process Zone Figure 7.17 A snapshot of failed elements (shown in black) in the results of a cellular automaton model with local rules of hardening and softening. (After Henderson et al., 1994.) White areas represent intact rock, stippled areas represent zones of damage containing isolated microfractures (generated by a hardening rule), and large black areas represent microcracks that have coalesced into a macroscopic fracture (generated by a softening rule). The two rules are combined in a single operator f, which modifies the fracture { toughness [ field Kc i,j ]} fkc i,j after the failure of a neighboring element, with f = 1 + ρ exp (n 4) 2 /20 e n2 /16, where n is the number of failed neighboring elements and ρ is an empirical negative feedback, or hardening, parameter. Large fractures are usually surrounded by a zone of damage similar to a process zone seen in ceramics and other composite materials. interactions then modify this spatial correlation, leading to a fracture pattern and cluster size distribution that both depend systematically on the relative degrees of hardening and softening in the mechanical constitutive rule. Figure 7.17 shows a high degree of spatial correlation, with large zones of undamaged intact rock, large correlated fractures, and clouds of microcracks both in isolated zones of damage and in process zones ahead of the edges of the larger fractures. The frequency-size distributions of the clusters (by area) has three regions (Figure 7.18). Region I is a steep slope for the smallest cracks, region II is a power law region with slope τ = 1 for intermediate cracks, and region III is an exponential tail that can be characterized by a finite correlation length. If the local hardening rule is absent, then models show only regions II and III (e.g., Bak and Tang, 1989). The transition between regions I and II has been seen in ceramic materials (Henderson

39 7.3 Mechanical and Structural Properties of Fracture Populations log N 5.0 p = 4.0 p = 2.0 p = Magnitude, m p = Figure 7.18 Equivalent cumulative frequency-magnitude distribution (log linear plot) for populations of synthetic cracks such as those shown in Figure 7.17, for different values of the hardening parameter ρ, also defined in the caption to Figure (After Henderson et al., 1994.) Increasing ρ implies greater local hardening. N is the number of earthquakes with magnitude greater than m the magnitude calculated from the logarithm of the area of clusters of connected failed elements. The data show three ranges: a characteristic peak at the smallest magnitudes, a scale-invariant range at intermediate magnitudes, and an exponential decay at high magnitudes. et al., 1994.) but the statistics are rarely sufficient to define the exponential tail of region III, for the very largest cracks, accurately. Nevertheless the inclusion of mechanical interactions (albeit in simple form) produces (in regions II and III) a frequency-size distribution identical to that predicted by the purely statistical percolation model [equation(7.78)]. The main difference is that the slope τ varies systematically with the feedback parameter in the physical model. As the stress concentration factor in the model is increased, τ decreases and ξ increases. This systematic behavior has been observed indirectly by monitoring acoustic emissions and measuring stress intensity factors in the double-torsion failure tests of Meredith and Atkinson (1983) and directly on the fracture patterns themselves by Hatton et al. (1993). Out-of-Plane Modeling Figure 7.19 shows the results of a conceptual model for fracture development in the form of normal faults by repeated earthquakes, after Cowie et al. (1993). The model lithospheric plate is stressed at its boundary at a constant strain rate, leading to local failure on the weakest elements in the form of model earthquakes, idealized as a step change in the local strain. This constant strain-rate boundary condition is a much more appropriate model for failure in the Earth, since the remote strain rate in the Earth is in fact remarkably constant (DeMets, 1995). Once individual earthquakes

40 400 Chapter 7 Hydromechanical Behavior of Fractured Rocks (a) (b) Figure 7.19 Fault patterns from the results of Cowie et al. (1993) at two different stages: (a) early and (b) later in the model run. Darker tones represent a greater degree of accumulated offset on the faults, normalized to take account of increased deformation on the two main strands [labeled 1 and 2 in (b)]. occur, the elements are assumed to regain their original strength in one time step. This is a reasonable assumption, since the healing time for individual faults, which has been estimated from the duration of aftershocks to be around a few months, is small compared with the hundreds or thousands of years for the average recurrence times for individual earthquake ruptures at a given location. The introduction of healing means that several cycles of global failure can be included, rather than the acceleration to a single failure for the tensile crack model of Henderson et al. (1994.). In parallel with local healing, the stress relaxed during the earthquake is redistributed according to all other elements by solving the equations of elasticity using a numerical scheme (in this example a conjugate gradient method). This is a significant advance on nearest-neighbor models, in that the spatial correlations that emerge are stronger and more realistic. The model of Cowie et al. (1993) was run without any preexisting spatial correlation in the heterogeneity of the starting material, leading initially to a random uniform distribution of nucleation sites, similar to the early stages of a bond percolation model (Figure 7.19a). However, the stress concentration and anisotropy of loading produced a highly directional growth and coalescence phase, leading to the development of large-scale faults in a braided pattern, with smaller faults in between having a proportionately smaller influence (Figure 7.19b). In fact, natural normal faults often occur within braided clusters, with concentrated zones of damage interspersed with relatively sparsely damaged zones (Shipton and Cowie, 2001). Such clustering and declustering of the fault pattern was not present in the starting model, and hence is purely a result of spatial correlations introduced by mechanical interactions in a heterogeneous composite medium. The smaller faults shielded from the remote stress field between the major faults retain a high

41 7.3 Mechanical and Structural Properties of Fracture Populations 401 stress level, and hence slip less often. This is consistent with geological observations of strain concentrating rapidly on the larger faults in the brittle regime. This primary feature is a necessary precondition for the development of sharp plate boundaries that is the first-order feature of global tectonics. Cowie et al. (1993) showed that this was an inevitable consequence of remotely accumulating strain on a heterogeneous material consisting of elastic-brittle elements. The frequency-size distribution follows the same form as equation(7.78), with an exponential dominance at the nucleation stages, a modified gamma distribution in the growth phase, and a broader band power law distribution at the percolation threshold (Cowie et al., 1995). The best-fitting slope for τ decreases from τ 1.8 to τ as the correlation length increases toward the percolation threshold and then remains relatively stable thereafter. The correlation length (measured by the number of broken bonds in the largest cluster, n max ) increases nonlinearly, accelerating through the growth and coalescence phases, and then stabilizes above the percolation threshold. This is consistent with simpler percolation and meanfield models. However, the model illustrated the fact that long-range physical (elastic) interactions are an essential ingredient to explain the spontaneous emergence of a realistic structure and pattern in the fault population. How do we quantify this complexity of structure? Correlations due to the concentration of strain can be quantified by calculating a fractal dimension D based on the space-filling properties of the objects but weighted by the local displacement to a power q (Cowie et al., 1995). For our purposes the multifractal spectrum D q corresponds to the qth moments of the spatial distribution of slip. Such a characterization is known as a multifractal spectrum, a concept that has been described as not for the faint hearted.a full description is given in Feder (1988), and the particular application to geological strain is illustrated in (Cowie et al., 1995) and Bonnet et al. (2001). In this characterization, the zeroth moment corresponds to the default capacity (space-filling) dimension, as used in equation(7.79). This is commonly measured by an unweighted box-counting method, as outlined in standard textbooks (e.g., Turcotte, 1992). The first moment, equivalent to the boxcounting algorithm but weighted by the local strain, is referred to as the information dimension (q = 1) and the second moment, weighted by the local strain squared, to the correlation dimension (q = 2). For a random uniform distribution of strain in two dimensions, D q = 2 for all q. In fact, different models produced by running the numerical model of Cowie et al. (1993) for different random initial-strength distributions all showed a capacity dimension D = D 0 = 2 that did not change with time. This is not surprising, since the capacity dimension is dominated by the presence of small cracks that nucleated at random early in the crack development phase. However, the higher moments progressively reduce through the growth and coalescence phase, with D 0 >D 1 >D 2. The reduced correlation dimension reflects the greater strain localization on and around the largest faults and the switching off of smaller faults in the stress shadows between major faults. This quantitative reduction in the correlation dimension, associated with an increase in clustering of large faults and a gradual concentration of slip on the largest faults,

42 402 Chapter 7 Hydromechanical Behavior of Fractured Rocks is a natural structural consequence of the mechanics of a growing and interacting fault population. What then of the scaling of displacement and length? Renshaw and Park (1997) used a boundary-element model without local healing, to simulate the growth of a tensile fracture population similar to that seen by Hatton et al. (1994) in the fissure swarms in volcanic basalts laid down in Northern Iceland. Fracture growth was assumed to occur quasi-statically according to Charles law, in response to stress applied at the sample boundary. The full elastic solution to the stress redistribution problem was solved using a numerical boundary element method, allowing crack crack interactions to be included accurately. This model successfully explained the scaling properties observed by Hatton et al. (1994), with u max l 2 for small lengths l and u max l for large lengths. This change of scaling occurred when fracture density was a critical fraction of fracture spacing. However, to fit the data quantitatively, the model required a much smaller value of n than normally associated with double-torsion tests on basalt in the laboratory tests described by Atkinson and Meredith (1987). However, smaller values of n have been inferred from laboratory acoustic emissions, where the damage was more distributed in compressive failure tests (Main et al., 1993). Thus, the value of n itself is not a material parameter but depends on the degree of stress concentration in the overall damaged material. In the double-torsion test the crack is nucleated at a notch and grows along a precut groove to minimize the size of the process zone, and hence then sample size, that is required. A large population of cracks will have a process zone comparable to the sample size, with a much lower stress intensity at individual crack tips due to stress shielding by the other cracks in the population. This results in more stable rheology that not surprisingly translates into lower effective values of n for the population as a whole. In summary, numerical models reproduce all of the scaling features seen in simpler mean-field models but add the important dimension of spatial correlation and related effects such as systematic changes in the evolution of the lengthdisplacement relation. They produce power law scaling rules for the fracture population that have the same form as mean-field models, but with different exponents such as τ and ν, and introduce new ones, such as the correlation dimension, that reflect the nonrandom spatial distribution of a set of growing interacting cracks. All of these structural properties of fracture populations strongly affect the hydraulic properties. Before we discuss these, we describe the actual scaling properties of natural fractures Scaling Properties of Faults and Fractures from Observations In this section we address the field evidence for such scaling in natural and experimental fracture sets. Natural fracture sets give us a picture of the total sum of the evolution of the fracture pattern, whereas experiments add the dimension of

43 7.3 Mechanical and Structural Properties of Fracture Populations 403 time, at the expense (like the numerical models) of simplifying the material and boundary conditions considerably compared with the Earth. Analogue Models It is useful to begin with results from an analogue model. Mansfield and Cartwright (2001) constructed a sandbox experiment containing a layered sequence of plaster and barite mud for normal faulting. The boundary conditions of constant strain rate were identical to those of Cowie et al. (1993), except that the analogue model is three dimensional. The model was analyzed by photographic time sequences to elucidate the evolution of the fault pattern. The faults grow in very much the same way as the model of Cowie et al. (1993), accumulating displacement during fracture growth, with deformation localizing on individual faults and fault clusters and large areas of low strain between. The displacement profile showed a very irregular distribution, associated with the complex pattern of linkage and growth during the history of slip. Each fault is a superposition of many previous smaller faults, leading to a rough displacement profile but also to a maximum displacement versus length relation that has a large and irreducible scatter that is also observed for tensile fractures in the field (Hatton et al., 1993). Thus faults may accumulate displacement without growing significantly and then suddenly increase their length by fault linkage, without adding significant displacement. The frequency-length distribution shows the same broad pattern of reducing τ and increasing ξ in the evolution to failure, as shown in the numerical models previously described. For this model, the power law component of the size-frequency distribution refers to the length of the fracture exposed at the surface: n(l)dl = (l/l 0 ) a dl, (7.91) where n is a density. The actual value of the slope on a log-log plot of frequency and magnitude depends on whether the data are binned linearly or logarithmically, so the density exponent is the more fundamental (Bonnet et al., 2001). To remove this ambiguity, we refer only to the density exponent a hereafter. The frequency-size distributions in the analogue model are in fact much more bumpy than equation(7.78). Mansfield and Cartwright (2001) point out that such bumps do not occur at consistent length scales for all the experiments and hence are not characteristic sizes introduced by the sample geometry. Instead, they are more likely to be due to statistical sampling effects that would disappear if the average of several experiments were taken. In fact, with a small number of faults, it is very easy to produce apparently significant bumps in the frequency-magnitude distribution from random statistical fluctuations, especially when analyzing cumulative frequency data (Main, 2000). In summary, the broad picture of nucleation, growth, and linkage and their effect on the size-frequency distribution, is very consistent between analogue and numerical models, but the analogue models produce a greater stochastic scatter about the best-fitting scaling relations. This irreducible scatter makes accurate prediction of individual cases very difficult.

44 404 Chapter 7 Hydromechanical Behavior of Fractured Rocks On a larger scale, the deformation of the Earth s lithosphere involves the response of a brittle upper layer overlying a ductile substrate. The presence of a ductile substrate allows a range of gravitationally driven processes to influence the pattern of growth of faults in the brittle layer. It also introduces a more homogeneous strain at the lower boundary of the brittle layer, in contrast to numerical experiments that consider forces only at the sample edges. Sornette et al. (1993) used a layered model consisting of an upper brittle layer of sand overlying a ductile layer of silicone putty. The silicone putty was chosen because it applies a nearly uniform strain on the underside of the brittle layer, in response to strain at the boundaries. The resulting deformation is more distributed than that of the numerical model of Cowie et al. (1993), and is more similar to that seen in the continent continent collision. As in Cowie et al. (1993), they found that the exponent of the length-frequency distribution decreased through the test. In a variety of analogue experiments, the exponent approached a stable value of a = 2.0 for large strains. However, Davy et al. (1995) and Bonnet (1997) showed that the power law behavior was not universal. In models that produced distributed deformation (nucleation dominated), the length-frequency distribution was exponential, whereas very localized deformation (coalescence dominated) resulted in power law distributions. In the general case, for intermediate localization, the distribution has the form of the modified gamma law (7.78). Thus, the form of the frequencylength distribution is exactly as expected from the much simpler percolation model, although the values of the exponents are different. The effect of the mechanical interactions then is to change only the model parameters, rather than the analytical form of the model itself. The importance of localization is also revealed in experiments on intact rock samples. For example, Lockner et al. (1991) showed that small microcracks tend to cluster around larger ones during the failure of rock samples in compression. This is also observed in field exposures (Anders and Wiltschko, 1994), confirming that clustering of deformation in a damage zone around larger fractures is a key aspect of fracture geometry. Such localization is associated with the development of power laws in the frequency-length distribution (Krantz, 1983; Moore and Lockner, 1995). Thus, power laws and localization (clustering) appear to be strongly interconnected. These results are not confined to compressive stress fields and in fact are easier to demonstrate under controlled conditions in tensile tests. For example, Hatton et al. (1993) examined fracture patterns developed on doubletorsion test samples fractured at constant stress intensity and found a systematic negative correlation between the power law exponent and the stress intensity. The double-torsion test has a preferential nucleation surface that is open to observation during the test, unlike the experimental arrangement for compressive failure. This surface was polished before the test, so that new cracks showed up when the sample was lit at a low angle of incidence. The length distribution exponent varied in the range 2 <a<2.7 and reached a value of 2 when the stress intensity equaled the fracture toughness K c, irrespective of material or chemical environment. In these experiments, increased stress concentration also leads to a power law with a lower

45 7.3 Mechanical and Structural Properties of Fracture Populations 405 slope and a damage zone that is more concentrated around the main fracture. At low K, the power law exponent is strongly dependent on the presence or absence of a chemically active pore fluid, in this case water. When the sample is immersed in water at low stress concentration, the deformation tends to be dominated by nucleation of a distributed array of small cracks, with a higher power law exponent for the length distribution. Thus, a mechanical parameter, the stress intensity, controls a structural one, the length distribution. Empirically, the data can be fitted by an equation of the form ) a = 2 + m (1 KKc, (7.92) where the slope m increases systematically with the chemical activity of the pore fluid. As the stress concentration increases (K K c ), a 2, irrespective of m. Since stress concentration is also associated with damage localization and clustering, it is not surprising that a is also systematically correlated to the degree of localization of damage not just in these tests, but also more generally. Field Observations Although laboratory tests under controlled conditions provide a necessary check on the mechanical models that have been developed, the true test of their applicability is direct observation of the natural experiment that the Earth is conducting. However, there are significant pitfalls in the analysis of natural data. Bonnet et al. (2001) recently conducted an extensive, critical review on the scaling of fracture systems in geological media and compiled the available data on a range of the parameters previously described. In this section we highlight some of the main results. A simple result that can be obtained by plotting the different forms of statistical distribution (exponential, power law and gamma) is that it is necessary to have a large bandwidth of observation scales to discriminate between the different possibilities. For example, typically 2 orders of magnitude of observations between the minimum and maximum fracture length would be required to distinguish between the exponential and power law models. Quite often in the literature, one model is simply preferred a priori and no attempt is made to propose a competing hypothesis. Figure 7.20 shows a compilation from the literature of the length range of observation and the magnitude of the length density exponents a produced for a variety of fracture types: faults, joints, veins, and shear bands. Only a small minority of these studies in fact have a length range of 2 orders of magnitude and many have only half an order of magnitude. This calls into question both the applicability of the power law and the reliability of the quantitative values for a produced. As a consequence, there is a large scatter in the reported values of a, between 0.8 and 3.2. This scatter cannot be explained by changes in physical conditions such as equation(7.92) and is much more likely to be due to errors introduced by

46 406 Chapter 7 Hydromechanical Behavior of Fractured Rocks Measured Density Exponent, a (a) Core, Thin Section and Micro Outcrop Scale 1 N f Outcrop Scale Length Range (meters) Aerial Photo, Seismic and Sattelite Scale Faults Joints Veins (b) Exponent a Number Figure 7.20 Length density exponents a from the literature for samples with more than 200 fractures. (a) Each density exponent is plotted with respect to the range over which it has been determined. (b) Histogram of length exponent values. The total range of exponents has narrowed to and shows a more distinct maximum at around undersampling of the data, specifically the narrow bandwidth of most of the studies. In fact, all of the well-determined values for a are within the range of , i.e., the same range as the laboratory experiments of Hatton et al. (1993) and consistent with equation(7.92). The mean value from the compilation (calculated from the histogram shown on their Figure 11) is 2.27 and the standard deviation is Given this large statistical scatter introduced by the narrow bandwidth of observation, it is not possible to distinguish between the different types of fractures from the length distribution exponent alone, although such a systematic correlation with deformation style cannot be ruled out either. A major conclusion from the review of Bonnet et al. (2001) is that more broadband scaling studies are needed before assigning any physical meaning to the measured scaling exponents. In the meantime, narrow-band determinations of scaling exponents should be treated with some caution in modeling mechanical, structural or hydraulic properties. Bonnet et al. (2001) also address the problem of reconstructing three-dimensional scaling properties from one-or two-dimensional sampling (typically available from boreholes and field mapping, respectively). For an ideal isotropic fractal object this scaling is trivial (Mandelbrot, 1982) and takes the form D 3d = D 2d + 1 = D 1d + 2, (7.93) i.e., each time adding a dimension of unity as we go from one-to two-to threedimensional sampling. Here D is the fractal (capacity) dimension for q = 0. We have already seen, already for the site-percolation model, that D 3d = 2.53 and D 2d = 1.89, so this equation does not hold in the general case, even for the simplest model that produces fractal scaling. In the general case,

47 7.3 Mechanical and Structural Properties of Fracture Populations 407 D 3d = D 2d + B : B<1, (7.94) where the constant B = 0.64 holds for the percolation model. It is also quite common to extend the argument from equation(7.93) to assuming a similar scaling relation has been applied to the scaling exponent a (e.g., Marrett, 1996). In fact, the exponent a is not strictly a fractal dimension according to Mandelbrot s (1982) definition, since it contains no spatial information itself, although we have seen in the foregoing discussion that its value is correlated with the degree of clustering. It is however, included in Turcotte s (1992) more general definition of a fractal dimension. The experiments of Hatton et al. (1993) allowed a test of the hypothesis that an equation similar to (7.94) applies to the exponent a from experimental fracture sets. Here, the two-dimensional length distribution was observed on the rock sample surface, and the three-dimensional version inferred from recorded acoustic emissions, assuming a dislocation model for the seismic source. They found a positive correlation of the form a 3d = Aa 2d + B, (7.95) where A = 1.28 and B = 0.05 (after correction for the fact that Hatton et al. (1993) quoted exponents for the cumulative frequency exponents, not the density distribution). However, equation(7.94) could describe the data equally well. equation(7.93) was ruled out by the data. On physical and empirical grounds, then, fracture patterns are in general not isotropic fractal sets and cannot be treated as such. The most difficult aspect of fracture statistics to model is their spatial correlation. Bonnet et al. (2001) discuss this in some detail. They conclude that the best way of describing the spatial correlation is not to use the multifractal spectrum D q, but to use the correlation dimension, D c, defined by the correlation integral C(r) = 1 r n(r < R)dR. (7.96) N(N 1) r min Here, R is a dummy variable, r is the distance between the center of mass of a pair of faults, r min is the minimum value of r, n is the number of lines of length less than r between fault pairs, and N is the total number of faults. For a scale-invariant fracture spacing, C(r) r D c. (7.97) The correlation dimension was first used by Grassberger and Procaccia (1983) in the analysis of the phase space of chaotic systems to measure the strangeness of strange attractors. More recently, it has also found utility in quantifying the clustering properties of earthquakes (Main, 1996) and fractures (Bour and Davy, 1998, 1999) in real space. Empirically, the correlation dimension is found to be more stable than estimates of the mass fractal dimension or, specifically, the multifractal

48 408 Chapter 7 Hydromechanical Behavior of Fractured Rocks correlation dimension D 2. Although D c and D 2 should give similar information about the clustering properties, D c is often a more stable parameter (Bonnet et al., 2001, see their Figure 8). The utility of the fractal dimension and the correlation length can be illustrated by considering the spatial evolution of the percolation and physical models on a two-dimensional space of size L L. In the early stages of damage, we would expect a nucleation-dominated pattern, with many small fractures arranged randomly in space, so that ξ/l 1 and D c = 2.0. As time goes on, ξ/l increases, and a power law distribution results. At the same time, the deformation becomes less distributed and more localized, leading to a decrease in D c, tending toward 1.0 as deformation localizes more and more on a single large one-dimensional fracture. The numerical experiments of Cowie et al. (1995), who estimated the correlation dimension with the multifractal method, confirm that there is a systematic negative correlation between the correlation length and the correlation dimension. However, there is no one-to-one relationship as the model parameters and initial heterogeneity are varied. Thus any attempt at reconstructing a fracture pattern solely from scaling parameters must take into account this large, irreducible variability. Figure 7.21 shows the reported values of fractal dimension, using different methods to determine D, after Bonnet et al. (2001), and restricted to the theoretical range 1 D 2. The frequency distribution is peaked at D = 1.5 and D = Faults Joints Undetermined Fractal Dimension Exponents (a) Linear Size of the System (meters) 1.0 (b) Fractal Dimension Figure 7.21 (a) Compilation of fractal dimensions reported in the literature as a function of the approximate linear size of the fracture networks, after Bonnet et al. (2001). Only values measured on two-dimensional fracture networks have been reported. Reported values smaller than 1 and larger than 2 have been excluded. Note that fault networks are sampled mainly at kilometer scale, while joint networks are sampled mainly at outcrop scale. (b) Histogram of fractal dimensions. Although there is a large scatter, the histogram seems to show two peaks at D = 2.0 (nonfractal patterns) and D = 1.5.

49 7.4 Reconstructions of fracture patterns 409 The latter is consistent with distributed deformation dominated by a random nucleation process but may also reflect a bias introduced by the theoretical constraint D 2. The diagram shows that there is a distinction between joints (fractures with measurable shear offset) and faults (fractures with measurable shear offset). Faults occur on all scales, but joint studies are typically limited to less than 100 m or so. Also, joints tend to have higher D on average than shear faults, implying that their deformation style is more distributed and faults more localized compared with the average of all fracture types. In summary, there is significant overlap in the degree of spatial organization among fractures of different types, but there are trends that can be made out. Thus differences in the mechanical cause of fractures lead to differences in the degree of localization of the resulting deformation. Bonnet et al. (2001) discuss many of the problems associated with the charactarization of natural fracture patterns. These include the narrow bandwidth of the scale range of observation, the need to test non power law forms of the scaling relations, the need to improve the quantification of fracture clustering in space due to localization, the dimensionality of measurement, and the control of fracture type on the resulting pattern. Of these, perhaps the most important is the continuing need for large scale, broad-band studies over as wide a scale range as possible. 7.4 Reconstructions Of Fracture Patterns And Implications For Fluid Flow And Chemical Transport One of the main aims of the large effort to characterize fracture patterns empirically, or understand them mechanically, is to enable the prediction of hydraulic properties on a larger scale. At a fundamental level this requires (a) knowing the properties of an individual fracture, as covered earlier in this chapter and (b) knowing the structural properties of the fracture population, as covered in the previous section. Ideally the structural properties would be evaluated by running a variety of mechanical models for fracture evolution, each conditioned to have the same output relations for size, clustering, and orientation as a natural analogue. This could be expensive in terms of computational time. Alternatively, this might be done heuristically, by producing random synthetic realizations using a statistical model with the same geometrical attributes as the natural pattern. It is relatively easy to do this by randomly placing fractures in space, constrained by a given overall fracture density and frequency distributions of size and orientation. However, the resulting synthetic realizations often do not look realistic (Odling, 1992), mainly because the correlation between fractures a fundamental feature of their evolution as we have seen above is neglected. (Bour and Davy, 1999) attempted to include this spatial dimension using D c to define the clustering properties of fractures. Figure 7.22 shows an example from their paper. In this reconstruction, a set of N midpoints for each fracture is first thrown down like a set of seeds but

50 410 Chapter 7 Hydromechanical Behavior of Fractured Rocks (a) (b) (c) Figure 7.22 Fracture networks with the same length exponent (a = 2.7) and different dimension Dcorrelation: (a) D c = 2, (b) D c = 1.75, and (c) D c = 1.5. The number of fractures in each set is The method used for generating a synthetic fractal network is described by Bour and Davy (1999). conditioned to have a particular value of D c and fracture density n = N/L 2. Then, for each point, a length is chosen from a power law size-frequency distribution, implicitly assuming that the correlation length is much larger than the scale of observation. Finally, the orientation of the fault is chosen from a parent distribution, here peaked at two angles representing a set of conjugate fractures intersecting at around 60. One single realization may not be sufficient to cover the scatter in the resulting patterns, so it is advisable to carry out several runs using different starting seeds. Once the synthetic pattern or set of patterns has been produced, the largescale properties of fluid flow can be determined if the permeability of the fractures and the matrix rock (the nonfractured part) can be estimated. To effect the latter requires a formal relationship first between fracture length and displacement and then between displacement amplitude and permeability, using results similar to those presented at the beginning of this chapter for single fractures. Note that this relationship depends fundamentally on fracture type (shear fault, tensile fracture, or joint). All initially will show an increase due to dilatant displacement. Tensile fractures formed by remote boundary stresses will continue to open as the fault grows (e.g., Hatton et al., 1993), whereas joints grow by relaxation of internal stresses, with no visible increase in opening or shear displacement. Shear faults have a decreasing permeability to flow across the fault because of the production of poorly sorted fault gouge, although they may be episodically permeable along their dip and strike because of dynamic dilatant slip at the moment of failure. Lowporosity crystalline rock will generally show a larger relative permeability increase in the dilatant phase than high-porosity sedimentary rocks, which can even show a decreased permeability before the development of the first localized fault due to pore collapse (Wong and Zhu, 1999; Main et al., 2001). Fracture type and starting porosity have a strong first-order effect on the local permeability. The hydraulic properties can be predicted at a larger scale and then perhaps tested against data from well tests for fluid pressure diffusion or tracer tests for