2.1.4. The Power Exponent of Permeability

Prior to the invention of FS methods, it was not possible to image fracture permeability directly to determine its location and spatial distribution. However, permeability proxies such as the mineral alteration caused by fluid flow through fractures and the size distribution of microearthquakes, can be used to infer the power exponent that characterizes the spatial distribution of fractures in rock [11]. The permeability power exponent can also be inferred from Equation (2) for a pink noise spatial variation in porosity or by the direct field mapping of fractures.

Figure 5 shows how the square of the amplitude (spectral density) of well-log measurements of fracture density, porosity, and fluid-flow-related mineral alteration fall off as the spatial scale of measurement (frequency) decreases. The variation in signal strength with depth, measured by well-logs, is Fourier transformed. The logarithm of the square of the amplitude (fluctuation power of the signal, *S(k)*) of each Fourier component is plotted against the logarithm of the spatial frequency, *k*, of the component measured in cycles/km (or any other units of measure, since the log slope is not affected by the units). The spectral density method is well described by Malamud and Turcotte (1999) [23]. All the properties related to fluid flow, plotted in Figure 5, show approximately a pink noise, scale invariant, 1/k power distribution.

**Figure 5.** Time-series analysis well-log properties related to flow in fractures. From **top left to bottom right,** these properties are: acoustic velocity, gamma ray intensity, potassium abundance,potassium-thorium abundance, neutron porosity, thorium abundance, uranium abundance, mass density correlation, and mass density. In all these panels, the spectral density S(k) (y-axis) falls off with frequency as ~1/k, where k is spatial scale-length measured in number of cycles per km (x-axis).Data are from a sand-shale well-log provided to us courtesy of R Slatt [21,22]. The power-scaling exponent is shown in the red box at the upper right of each plot.

### 2.1.5. The Flow Significance of the Distribution of Permeability

Figure 5 shows that down hole log measurements related to flow in fractures are power-scaling with an exponent of approximately −1:

$$S(k) \propto 1 \emptyset^{\beta}, \text{ where } \beta = 1. \tag{4}$$

This value of the power exponent, β*,* is intermediate between zero slope of randomly distributed white noise and the slope of two of red noise (the Brownian noise, obtained by taking the running sum of randomly distributed spatial properties). A power spectrum with a slope of one is called a pink noise. Equation (4) shows that, for pink noise, when *k* becomes smaller (e.g., the number of Fourier cycles per km is fewer and the variation extends over a larger distance), the signal power increases. For a pink noise spatial distribution of permeability, as the volume of rock becomes larger, it is increasingly likely that a permeable channel will short-circuit flow across the entire observed volume. We show this below by first discussing how white and pink noise distributions of porosity di ffer, and then use Equation (2) to convert a pink noise porosity distribution to a pink noise permeability distribution and address channeling. 

Figure 6 contrasts white and pink noise distributions in porosity. The plots on the left side of Figure 6 depict a hypothetical white noise porosity distribution. The top image is a depth slice map of porosity. The middle plots the porosity along an arbitrarily directed hypothetical well-log through the depth slice map shown above. The Fourier power spectrum *S(k)* of this porosity well-log is plotted as a function of wave number *k* on the abscissa in the bottom image. The white noise is generated from a standard pseudo-random number generator in Matlab computational software.

**Figure 6.** (**a**): Spatially uncorrelated and (**b**) correlated hypothetical porosity models. (**Top**): Map views of porosity variation indicated by color. (**Middle**) Representative synthetic porosity well-logs in an arbitrary direction through the porosity map view images above. The ordinate is distance and the abscissa is the magnitude of porosity. (**Bottom**): Power spectrum of synthetic well-log porosity, plotting the square of harmonic amplitude (ordinate) against spatial frequency (cycles per km) on abscissa. Numbers are the power exponents (-β ) of the power spectra (see Equation (4)). They confirm that the left map distribution is spatially uncorrelated white noise (β ~ 0) and the right map distribution is spatially correlated 1/*k* pink noise (β ~ 1).

The working assumption behind the white noise application to crustal randomness is that, at some suitably large scale-length, whatever is physically happening at one point in the formation is independent of whatever is happening at any other point. That is, above a suitable scale length, crustal flow structure is uncorrelated. The white noise uncorrelation assumption is inherent in the

representative elementary volume (REV) method of crust flow modelling (e.g., [24–26]). Whatever the pattern in the REV, it is assumed that one could make a 3D rubber stamp of the REV and replicate the pattern at a larger scale by filling in the whole volume with the stamp. The largest and smallest features remain the same, no matter how many times a pattern is repeated or how large its volume is. REV patterns cannot increase or decrease the range of properties beyond those in the REV. REV's thus have no scaling larger or smaller than that contained within the REV itself. As a consequence, the entire crustal flow volume is described by just one REV sub volume. There is no large-scale variation in permeability in the depth slice white noise map (top right).

In contrast, the *1*/*k* spectrum pink noise-correlated spatial distribution of porosity is illustrated on the right side of Figure 6. The pink noise distribution of porosity is generated using discrete Fourier transform methods, as described in ([23], p.35). The pink noise porosity distribution is then shown along a hypothetical well-log through the depth slice image (middle right diagram) and plotted in power spectrum form in the bottom right image. The pink noise porosity distribution reveals di fferent intensities at every scale length contained in the volume. The plan map shows dramatic inhomogeneities which connect over large distances. A yellow to red band across the entire property map can be seen in the middle of Figure 6, for example. This porosity pattern suggests that a flow channel could pass fluid across the entire image. In contrast to the uncorrelated plan map depiction to the left, no small portion of the correlated map is representative of any other small portion. The porosity level and pattern at one location cannot be used to predict what will be found in another small portion of the image. The spatial fluctuation power spectrum has a power slope of ~1 rather than ~0. Connected flow paths are expected, but the position of the spatially correlated pore-connectivity cannot be predicted on the basis of a limited number of samples.

The tendency towards channelized flow at large scale for a *1*/*k* pink noise porosity distribution is demonstrated in Figure 7. Flow in this figure is calculated for a porosity distribution derived from a normal distribution of pore sizes with a *1*/*k* power spatially correlated distribution with a power scaling exponent of −1, in the same fashion as described above. The porosity normal distribution is converted to permeability using Equation (2), a pressure gradient is imposed across the spatial permeability distribution, and the resulting flow is computed using the Matlab finite-element partial di fferential equation solver. The 2D grid has 300 × 300 element resolution. Arrows show both the distribution of velocity magnitudes and flow directions in the model domain. The ±2 spread in the natural logarithm of the flow velocity (ln|V|) in Figure 7 approximately matches the spread of natural log permeability shown in Figures 2–4. The flow directions show that spatially correlated porosity leads to flow dramatically channeled across the model from the high-pressure boundary to the low-pressure boundary.

**Figure 7. a**. The flow field, computed for a medium with a permeability that varies by ±2 natural log units (see Figures 2–4 well-core data) and has a 1/k spatial distribution of ln κ, shows dramatic channeling between the high- and low-pressure boundaries. **b**. Distribution of natural log-well log-flow velocity, ln (V), where V is in arbitrary units

### 2.1.6. The Power Exponent of Permeability from Two-Point Analysis of Permeability

Figures 2–4 sugges<sup>t</sup> a mean value of φ = 0.2 ± 0.1. Assuming a critical state crust with αφ*o* = 4, it follows that α = 20, and, since α = σln <sup>κ</sup>/σϕ, σln κ<sup>=</sup> 2, then whether ln κ has a *1*/*k* crustal scaling can be tested by two-point spatial correlation methods. This analysis shows a power law separation exponent of −0.5. This corresponds to a Fourier power spectrum slope of *1*/*k* for the ln permeability field [10,27].

### 2.1.7. Power Law Exponent of Permeability from Microearthquake Distributions

Over 18,000 m<sup>3</sup> of water was recently injected at ~6 km depth in Finland to stimulate flow for deep geothermal heat extraction [28]. Over 54,000 microearthquakes (−1 < M < 1.9) resulted from this stimulation, of which 6150 were located and characterized. These events have a two-point correlation power law separation radius exponent of between −0.5 and −0.6 [10], which corresponds to a Fourier power spectrum slope of *1*/*k*.

### 2.1.8. Power Law Exponent from Field Mapping of Fractures

Barton (1995) [11] reported on outcrop fractures mapped at 17 locations at scales from centimeters to kilometers. He found that fracture length had a fractal dimension of *D*~1.5.(range 1.4 and-1.7) Since fractal dimension is related to the exponent β as 2β = *5-2D* ([23], p30), Barton found β ~ 1 from outcrop mapping. A β = 1 corresponds to a *1*/*k* distribution, since, by convention, β is the negative of the exponent of k.

### 2.1.9. Power Exponent from Analysis of Fracture Seismic Images

The flow backbone shown in the bottom image of Figure 1 consists of connected linear segments with different levels of fracture seismic emission intensity and physical length. Histograms can be made of both relative fracture seismic segmen<sup>t</sup> length and FS emissions intensity, and their statistical distributions determined. Figure 8 shows that both of these properties have lognormal (or power law, since the roll over may simply reflect the limits of segmen<sup>t</sup> resolution) distributions, in which there are long fracture seismic intensity segments, and high FS emissions intensities that fall far outside of normal distributions, i.e., these distributions have long tails.

**Figure 8.** (**Left**) Normalized length distributions of linear fracture seismic intensity segments from four different reservoirs: (Clockwise) Marcellus, Utica, East Asia, Wayland. Linear backbone segments can be seen in the fracture seismic map slice in the bottom image in Figure 1. (**Right**) Normalized histograms of seismic fracture emission intensity recorded shale reservoir fracks, above. Data are from Lacazette et al. (2013) [29].

### *2.2. Flow From a Well: What Is at Stake*

Figure 9 illustrates what is at stake in practical terms in the description of permeability as both lognormal and *1*/*k* distributed. The figure depicts flow from a well to the sides of the computation domain, where the injected fluid is allowed to freely escape the system. The figure was generated by Matlab finite-element methods for different values of β and α in Equations (2) and (4). Uncorrelated (β = 0) and correlated (β = 1) porosity distributions are assigned to 10<sup>5</sup> unstructured elements with the grid resolution increasing strongly toward the injection well. Corresponding permeability was derived from porosity assuming α = 1 or 30. The mean porosity is selected such that αφ0 ~3 to 4 in all panels. The method of flow computation is the same as is used to simulate the flow vectors shown in Figure 7.

If the permeability has little variation around its mean (small σln κ), the flow proceeds radially away from the well (lower two images) regardless of whether the ln permeability (and porosity) is spatially ordered as pink (β =*1, 1*/*k*) or white (β =*1,* random) noise. However, if the magnitude of permeability variation around its mean is of the order shown in Figures 2–4, and the ln permeability is spatially ordered (*1*/*k*), flow from the well occurs through a spatially correlated dendritic or filamentary pattern (top left image). The flow does not have this pattern if the ln permeability distribution is uncorrelated (white noise), even if the permeability has a large spread from its mean (top right image). Both substantial permeability inhomogeneity and a pink noise spatial ln permeability distribution are required to produce the channelized/filamentary flow shown in the top left panel of Figure 9.

**Figure 9.** Flow calculated from a central well to free-flow boundaries where it escapes the system for different values of α in Equations (2) and (3) and β in Equation (4). Color indicates the relative magnitude of the fluid velocity (red low, purple high). Flow channeling (**upper left panel**) requires that both a large spread in ln κ from its mean (e.g., α =30) and a distribution in permeability that is pink (β = 1). The variation in the magnitude of permeability from its mean can be large, but cannot produce channel flow if its spatial variation is that of spatially uncorrelated white noise (**top right panel**). The permeability distribution can be 1/k (pink noise) but cannot produce channel flow unless the variation in ln permeability is also large (**bottom left panel**). When the permeability distribution is white and the spread in ln permeability small, the flow from the well is of course uniform (**bottom right panel**).

The earth's crust meets both these criteria for channelized flow (permeability inhomogeneity and pink noise, scale invariant, spatial ln permeability distribution) as the following references in various areas attest: oil/gas (USEIA 2011), geothermal [30], mineral deposition [31–36] and trace elements [37–40]. The spatially heterogeneous, length-scale dependence of permeability is documented from lab-scale cm to reservoir-scale km [8,41–45]. The normal distributions, often assumed for modeling and analysis of flow in the earth's crust, o ffer no formalism for dealing with the spatial variations (channeling) observed in the actual flow system (e.g., [21,22,46–55]).
