**2. Crustal Observations and Implications**

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

This section first shows that spatial variations in crustal porosity invariably coincide with commensurate spatial variations in the natural logarithm of permeability. It then shows that this means

that the magnitude of crustal permeability commonly has a lognormal distribution, even though the magnitude of porosity that is spatially associated with permeability is normally distributed. Following Leary et al. (2017, 2018) [12,13], we derive a general expression that encapsulates this observation and deduction. The derivation shows that the combinatorial probability of crustal pore and fracture connections provides a logical explanation for why normally distributed pores are associated with a lognormal distribution of permeability. There could be a physical extension between pores and fractures [13] which we do not discuss here. With increasing strain, broken grain cements could progressively localize deformation, eventually leading to faulting, for example. This is not addressed in this paper. Here, we use observations of cores to discuss the connection conditions for permeability and learn how a normal distribution of pores and pore connections leads to the observed lognormal distribution of core permeability. We then transfer this insight to fractures.

Figures 2–4 show field data that illustrate how changes in crustal porosity are invariably associated with changes in the natural logarithm of permeability,

$$
\delta q \propto \delta \ln \kappa \tag{1}
$$

The (possibly irregularly spaced) porosity and permeability values in these figures were measured down well-core samples according to standard oil-field procedures (e.g., [14]). The histograms on the left side of Figure 2 show that the porosity values of four North Sea cores have a normal spatial distribution, whereas the permeability values have a lognormal distribution. Similar histograms of porosity and permeability are shown in Figures 3 and 4.

On the right side of Figure 2, the porosity and permeability values are plotted as a function of sample number. In Figures 3 and 4, they are plotted as a function of sample depth. The measurements are plotted in normalized form, such that their variations have zero-mean and unit variance. This is achieved by subtracting the mean porosity (or ln permeability) and dividing by the standard deviation of porosity (or ln permeability). These profiles show that where there is a change in porosity there is a correlated change in ln permeability. Natural logs are used throughout the discussion for consistency with these observational plots and for compatibility to theory.

Whereas the number of samples with small porosity values rapidly decreases in these histograms, the same is not true for permeability histograms. The roll-o ff in the number of samples with small permeability values can be seen only in the plot of numbers versus the logarithm of permeability, and may be due to failure to measure variations in low permeability. The potential for undersampling suggests that the permeability distribution could be alternatively described as a power law (which has no roll-o ff at low permeability) instead of lognormal. Both logarithmic and power distributions are long-tailed at the high permeability end of their distributions. However, power distributions are fat-tailed, meaning this distribution has more high permeability values than a lognormal distribution.

The important point here is not these di fferences between power and lognormal distributions, but simply that, on simple inspection, the porosity and permeability distributions are normal and lognormal, respectively, and these relationships are general. The Figures show data from wells worldwide. They depict the most commonly observed characteristic of reservoir rock porosity and permeability. Furthermore, the normal-versus-lognormal correlations between porosity and permeability are not dependent on the orientation of the sampling well. Figure 4 shows that the same relationship is observed in cores from a horizontal well as in cores from vertical wells. Be it a vertical or horizontal well-core sampling sequence, the di fferences in their population distributions remain the same: normally distributed porosity and long-tail power permeability. Both these distribution and correlation relationships appear to apply just as well along rock formations as across them.

**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 sugges<sup>t</sup> 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).

Figures 2–4 show data that establish Equation (1). Equation (1) can be integrated to yield an expression between the spatial variation of permeability, κ(*x,y,z*), and the spatial variation of porosity, ϕ (*x,y,z*)

$$
\kappa \approx \kappa\_0 \exp(\alpha(q - q \eta)),
\tag{2}
$$

**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.

Here, κ0 and ϕ0 are the mean values of permeability and porosity, and α is a constant of proportionality. That (2) follows from (1) can be easily verified by differentiating (2) with respect to ϕ. If ϕ = ϕ0 ± σϕ, it follows from (2) that ln κ = ln κ0 ± ασϕ, where α is the ratio of the standard deviation of ln κ to the standard deviation of ϕ:

$$
\alpha = \sigma\_{\ln k} / \sigma\_{\Psi}. \tag{3}
$$

When α is very small, the series expansion of the exponential in Equation (2) reduces to a linear relationship between the departure of κ and the departure of ϕ from their mean, and both permeability and porosity are normally distributed. In the small-α limit, permeability equations such as the Kozeny–Carmen relationship hold. In this domain, the flow is homogeneous for a normal distribution of porosity.

When α is large, κ fits the definition of a lognormal variable, where the permeability distribution has a long permeable tail. In the large-α limit, ln κ has a standard deviation spread of σln κ. If this spread is large enough and, as discussed in the next section, the spatial distribution of permeability is a power distribution, the flow will be channelized. These two conditions (large α and a power spatial distribution of permeability) for channelized fracture flow are what we investigate in this paper.

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

The reason that a normal distribution in porosity magnitude can lead to a lognormal distribution in permeability magnitude can be understood if it is recognized that permeability requires a connection condition to be met. For the current discussion, this connection condition is that fractures (or pores) must be able to pass fluids onward to other fractures (or pores).

Shockley (1957) [15], among others, illustrated how physical/statistical observables (events) that are normally distributed can interact conditionally to create other observables (events) that are lognormally distributed. His simple example asked why relatively few authors publish the bulk of scientific papers. He proposed that eight author traits are necessary to produce a publication: (1) ability to select problems; (2) competence to work on them; (3) ability to recognize a result; (4) ability to prepare a manuscript; (5) ability to present results adequately; (6) ability to profit from criticism; (7) determination to submit a manuscript; and (8) ability to respond to referees' criticism. Assuming each author trait is normally distributed through a population of scientists, the conditional nature of the final 'event', the publication of a scientific paper, depends on the product rather than the sum of the component traits. If p1 is the probability for trait 1, p2 for trait 2, etc., then *P* = *p1\* p2\** ... *\* p8* 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.

Permeability requires that pores and fractures be connected. As the scale or volume of interest becomes larger, there will be more and better connections between pores, and longer and more permeable fractures within a given volume, and the probability that some subset of pores or fractures will be connected will increase with increasing volume. For example, given n independent fractures in a volume, simple counting shows there are *n!* = *n\* (n-1) \* (n-2) \** ways to connect them. If the number of pores or fractures increases by δ*<sup>n</sup>*, the probability of connection will increase: δ*ln(n!)* = *ln([n*+δ*n]!)* − *ln(n!)* ≈ δ*n ln(n).* Since *ln(n)* changes more slowly than *n*, it is reasonable to assume that δ*ln(n!)* ∝ δ*n* Associating *n* with ϕ, and *n!* with κ*,* we find δ*ln(*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

Critical state physics principles can account for how variations in permeability become distributed spatially. This distribution is the second and final piece of information needed to determine whether the departures of ln permeability from its mean will lead to flow channeling.

One example is the application of critical state physics to the phenomenon of opalescence. Opalescence is the clouding of an otherwise clear liquid such as CO2 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.

Properties that vary spatially in a power law 1/k fashion are scale invariant in the sense that you cannot deduce scale in an image of the material that contains the property. In the case of opalescence in a clear liquid, the pattern of optical index variations captured in a photo frame of tenths of a millimeter on a side looks the same and is statistically the same as that captured in frames that are millimeters on a side. Furthermore, the onset of density variations and the clouding they cause is related to a "structure function" such as ε = *(T – Tc)*/*Tc*, where *T* is the system temperature, and *Tc* 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.

The critical-state percolation model [17,18] provides additional insight. In the simplest version of a percolation model, nodes in a square grid are connected to their neighbors by bonds that are either open (and allow flow between nodes) or closed (and allow no flow). If open bonds are randomly assigned according to some probability, there is a critical fraction of open bonds at which at least one pathway of conducting bonds spans the sample. Near this critical sample-spanning fraction, percolation model properties such as permeability, electrical conductivity, and di ffusion coe fficient are found to be power functions, and universally depend more on spatial dimension than the morphology (internal construction) of the medium [18]. This potentially greatly simplifies understanding rock properties. The flow backbone in the sample-spanning cluster is found by deleting all the dead-end conducting bonds where no flow occurs. The flow backbone provides a basis for the depiction of backbone fracture seismic flow structures shown in the bottom section of Figure 1. Note that the seemingly dead-end backbones of flow in Figure 1 could connect to overlying or underlying depth slices through the volume, or could discharge into a broadened flow.

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

A major insight of the late 20th 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 αϕ0 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 αϕ0 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 αϕ0 is between 3 and 4 for a wide variety of sedimentary rock, which means that they are in a state of incipient failure.
