# Observational and Critical State Physics Descriptions of Long-Range Flow Structures

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{ο}= 3 to 4 and that the natural log of permeability has a 1/k pink noise spatial distribution. Combined, these conclusions mean that channelized flow in the upper crust is expected as the distance traversed by flow increases. Locating the most permeable channels using Seismic Fracture methods, while filling in the less permeable parts of the modeled volume with the correct pink noise spatial distribution of permeability, will produce much more realistic models of subsurface flow.

## 1. Introduction

## 2. Crustal Observations and Implications

#### 2.1. The Observed Relationship between Crustal Porosity and Permeability

**κ**(x,y,z), and the spatial variation of porosity, φ (x,y,z)

#### 2.1.1. The Connection–Condition Explanation for a Lognormal Distribution in Permeability

_{1}is the probability for trait 1, p

_{2}for trait 2, etc., then P = p

_{1*}p

_{2*…*}p

_{8}is the probability of publication, and only a few authors will possess all the traits needed for successful publication. Successful authors will be rare, and the distribution of author publications will have a long tail: few scientists meet the connection condition of having all eight traits, so they publish a disproportionate share of scientific papers.

_{*}(n-1)

_{*}(n-2)

_{*}ways to connect them. If the number of pores or fractures increases by δn, the probability of connection will increase: δln(n!) = ln([n+δn]!) − ln(n!) ≈ δn ln(n). Since ln(n) changes more slowly than n, it is reasonable to assume that δln(n!) ∝ δn Associating n with φ, and n! with κ, we find δln(κ)∝ δφ, as shown in Equation (1), above. We thus see that, if permeability is viewed in terms of connections of pores or of fractures, it follows that a change in the number of pores (pore connections) or in the number of fractures will lead to a change in ln permeability.

#### 2.1.2. Spatial Properties at the Critical State

_{2}at its critical temperature and pressure. Observed almost 200 years ago, this phenomenon was understood 100 years later to be related to spatial optical index fluctuations in the critical liquid which scatter light, thereby clouding the liquid. In the last 50 years, the amplitudes of these fluctuations have been found to be power law distributed as a function of their physical length scale. The longer the length scale of the fluctuation, the larger the amplitude of its density change. For example, if the density variations were decomposed into their Fourier components, the longer wavelengths would have a greater amplitude. The square of the amplitude might vary as 1/k, where k=2π/ λ, where λ is the wavelength of the Fourier component.

_{c})/T

_{c}, where T is the system temperature, and T

_{c}is the critical point temperature (e.g., [16]). Importantly, the opalescence is clearly related to the proximity to the critical point and not to other properties of the fluid.

#### 2.1.3. The Critical State of the Earth’s Crust

^{th}Century in the Earth Sciences is that the earth’s crust is stressed by plate tectonic forces, such that it is in a state of constant incipient failure [19]. The brittle crust is riddled with fractures that are always at or near mechanical failure. Fractures are power scaling or scale-invariant characteristics of rock [11]. Geologists must place a hammer or some other object of known scale in a picture of veins or fracture traces to indicate their scale (e.g., [11]). Leary et al. (2018) [13] have shown that the αφ

_{0}parameter in Equation (2) above, plays the same role in the deformation of the earth’s crust as the critical temperature plays in opalescence. If αφ

_{0}is between 3 and 4, the crust is in a state of constant incipient failure. The result is that the spatial distribution of fractures is power law, and it follows that the spatial distribution of permeability is power law. Barton, Camerlo, and Bailey, 1997 [20] have shown that the distribution of fracture trace lengths is also power scaling. Leary et al. (2018) [13] show that αφ

_{0}is between 3 and 4 for a wide variety of sedimentary rock, which means that they are in a state of incipient failure.

#### 2.1.4. The Power Exponent of Permeability

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

^{β}, where β = 1.

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

_{o}= 4, it follows that α = 20, and, since $\alpha ={\sigma}_{\mathrm{ln}\kappa}/{\sigma}_{\phi}$, ${\sigma}_{\mathrm{ln}\kappa}$= 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

^{3}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

**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

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

^{5}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 αϕ

_{0}~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.

## 3. Summary and Discussion

_{ο}= 3 to 4 and permeability has a 1/k (pink noise) spatial distribution. Consequently flow in the crust will become increasingly channelized as the distance of the flow increases.

_{ο}= 3 to 4 defines the critical point of failure for the crust, for which this scale invariance will pertain.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**Top**) A horizontal map slice of the above background Fracture Seismic (FS) intensity (red high, blue low), activated during the third stage of a fracture stimulation treatment. (

**Bottom**) The above-background FS intensity (red high, blue low) backbone of the same horizontal map slice in the FS intensity cloud for hydrofracturing Stages 3–5. Point A is the wellhead of a horizontal well with a trajectory indicted by the black line. Point B shows the location of a vertical well with pressure and chemical tracer monitors. Both pressure and tracer communicated between Well B and Well A, where the FS skeleton connects the wells over a distance of 600 m. Figure is from Lacazette et al (2013) [3].

**Figure 2.**Porosity and permeability measurements on four sub-vertical North Sea gas field well cores. (

**Left panel**) Histograms of porosity values are plotted as number of values in 1% wide bins versus percent porosity. Both log permeability and permeability data are plotted as the number of values in a bin versus millidarcies. The histograms suggest that the porosity has a normal, and the permeability a lognormal, distribution. (

**Two right panels**) Porosity (blue) and natural ln permeability (red) data are plotted as functions of sample number. The porosity and permeability data are normalized to zero mean and unit variance.

**Figure 3.**Porosity (ϕ, blue ) and natural log permeability (ln κ, red) magnitudes in sub-vertical well cores from the North Sea (top), Germany (middle), and South Australia (bottom). Porosity and ln permeability are plotted against sample depth. Both are normalized to zero mean and unit variance. The number of measurements is shown on the plots. In the histograms, the porosity values are plotted as the number of values in 1% wide bins versus percent porosity. Both ln permeability and permeability data are plotted as the number of values in bins versus millidarcies (mD).

**Figure 4.**(

**Top**) Porosity (ϕ, blue) and ln permeability (ln κ, red) measurements from a horizontal well from North Sea clastic oil fields, plotted with zero-mean and unit variance. (

**Bottom**) Histograms of the number of measurements in bins of equal porosity, ln κ, or κ.

**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.

**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).

**Figure 7.**

**a**. The flow field, computed for a medium with a permeability that varies by ±2 natural log units (see Figure 2, Figure 3 and Figure 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

**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].

**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**).

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**MDPI and ACS Style**

Malin, P.E.; Leary, P.C.; Cathles, L.M.; Barton, C.C.
Observational and Critical State Physics Descriptions of Long-Range Flow Structures. *Geosciences* **2020**, *10*, 50.
https://doi.org/10.3390/geosciences10020050

**AMA Style**

Malin PE, Leary PC, Cathles LM, Barton CC.
Observational and Critical State Physics Descriptions of Long-Range Flow Structures. *Geosciences*. 2020; 10(2):50.
https://doi.org/10.3390/geosciences10020050

**Chicago/Turabian Style**

Malin, Peter E., Peter C. Leary, Lawrence M. Cathles, and Christopher C. Barton.
2020. "Observational and Critical State Physics Descriptions of Long-Range Flow Structures" *Geosciences* 10, no. 2: 50.
https://doi.org/10.3390/geosciences10020050