Sublinear scaling of fracture aperture versus length: An exception or the rule?

Abstract

[1] Observations of natural fracture dimensions have sparked a continuing debate as to the nature of the fundamental relationship between fracture aperture (maximum opening) and length. On the basis of theoretical fracture mechanics, some have argued aperture-tolength scaling should be linear. This relationship implies that all fractures in a given population have the same driving stress regardless of fracture length, arguably a state that is difficult to reconcile with fracture propagation criteria. Also, some field observations indicate sublinear aperture-to-length scaling that is apparently inconsistent with the linear elastic fracture mechanics theory. In this work, a nonlinear aperture-to-length relationship is derived, still based on linear elastic fracture mechanics in a homogeneous body, but incorporating subcritical and critical (equilibrium law) fracture propagation criteria. The new hypothesis postulates that fractures of different lengths preserved in a body of rock are all in the same condition with respect to propagation (i.e., they all have the same stress intensity factor). This requires that fractures have driving stresses that vary inversely with the square root of fracture length, producing fracture apertures that scale with length to the 1/2 power. Under these conditions, fracture aspect ratio (aperture/length) decreases with increasing fracture length to the negative 1/2 power. Linear aperture-to-length scaling is still considered a possibility but is attributed to a relaxed, postpropagation mechanical state. Deviations in fracture aperture-to-length relationships from these idealized models can result from mechanical fracture interaction, fracture segmentation into en echelon arrays, and three dimensional effects in stratabound fractures.

The impact of different aperture distribution models and critical stress criteria on equivalent permeability in fractured rocks

“…Although some studies note that vein [Olson, 2003;Hooker et al, 2012], there is little explicit evidence derived from outcropping veins. We model aperture-frequency distributions by assigning apertures independent from fracture geometry using power law exponents of 0.8 (the universal scaling exponent for apertures in sandstones), 0.7, and 2.0 (common values found for carbonate rocks; Figure 1a) [Hooker et al, 2014, and [Lawn and Wilshaw, 1975], where the maximum displacement d max (in m) in the direction perpendicular to the fracture walls is defined by…”

“…The subcritical toughness is generally defined as K Ã C ≅K C =10 [Atkinson, 1984;Segall, 1984;Olson et al, 2009]. The behavior of fracture toughness with respect to length defined by equation (2) is matter of debate, as K c is either considered to scale linearly with length on all scales, or only for small fractures [Olson, 2003;Scholz, 2010Scholz, , 2011Olson and Schultz, 2011;Anders et al, 2014]. In the case of a constant K Ã C , aperture scales with the square root of length (i.e., sublinear) instead of linearly [Olson, 2003].…”

“…The behavior of fracture toughness with respect to length defined by equation (2) is matter of debate, as K c is either considered to scale linearly with length on all scales, or only for small fractures [Olson, 2003;Scholz, 2010Scholz, , 2011Olson and Schultz, 2011;Anders et al, 2014]. In the case of a constant K Ã C , aperture scales with the square root of length (i.e., sublinear) instead of linearly [Olson, 2003]. Depending on the chosen parameters used to define the scaling behavior, the linear and sublinear trends may intersect.…”

“…Δσ I is the driving stress (in Pa), which is the difference between fluid pressure P f within the fracture and normal stress σ n acting on the fracture [Lawn and Wilshaw, 1975;Pollard and Segall, 1987;Vermilye and Scholz, 1995;Olson, 2003;Anders et al, 2014]. Critical crack growth occurs when P f > σ n , i.e., when hydraulic fracturing takes place [Segall, 1984;Engelder and Lacazette, 1990].…”

Influence of Landscape Coverage on Measuring Spatial and Length Properties of Rock Fracture Networks: Insights from Numerical Simulation

“…Thus, analysing the vegetation cover effects on the measurement of anisotropic fracture networks will be a focus of our future work. Furthermore, in the fracture network model, we represented fracture traces as 1D lines of the same width without taking into account the variation of fracture apertures, which may scale with fracture length following a power law (Vermilye and Scholz 1995;Renshaw and Park 1997;Bonnet et al 2001;Olson 2003;Neuman 2008). Very large faults with wide apertures/damage zone thickness tend to be more observable and less affected by landscape patches than small-scale fractures.…”