**3. Summary and Discussion**

Flow measurements are di fficult to make at even a few meters from a drill hole. However, we can infer the characteristics of flow far away from the wellbore from the observed variation in permeability about its mean and from the observed spatial distribution of permeability. Our analysis proceeds from the observations that (1) variations in crustal porosity and pore connections are invariably associated with variations in the logarithm of crustal permeability (δϕ ∝ δ ln κ) and (2) the Fourier spectral power fluctuations in permeability magnitude (and all rock properties, micro-earthquakes, etc., associated with flow in fractures) scale inversely with spatial frequency (*1*/*k*).

Integrating the first observation produces an exponential relationship κ ≈ κ0 exp(α(ϕ − ϕ0)), in which the variation in permeability magnitude relative to its mean is specified by a parameter α = σln <sup>κ</sup>/σϕ. The permeability in the Earth's crust has a substantial spread and the Earth's crust is critically stressed which means that αφ*o* = 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.

The relationships described in this paper are physically fundamental. A connection condition explains why changes in porosity or the number of fractures in a volume are related to changes in log permeability magnitude. The pink noise spatial distribution of permeability is the distribution for which scale cannot be determined from observations. Geologists have long been aware that geological observations are scale invariant [56,57]. The parameter combination αφ*o* = 3 to 4 defines the critical point of failure for the crust, for which this scale invariance will pertain.

Critical State Physics studies show that near the point of critical failure many parameters have a pink noise spatial distribution and that their magnitudes depend more on the proximity to critical failure than to other factors [18]. This greatly simplifies understanding crustal permeability. That fractures are closely involved in the critical state of the crust may not be surprising. Fluid-filled fractures are the weakest element in the crust. The crust may be in a critical state because, in this state, fluids can move, as required by compaction and tectonic deformation, through permeable pathways at all scales (millimeters to kilometers). Fluids may control crustal properties more than presently appreciated.

The framework described in this paper is also of grea<sup>t</sup> practical importance. Flow channelization poses a risk to many engineering and commercial pursuits. That flow channelization must be expected in the Earth's crust as the distance of flow increases is thus significant. The fact that flow channels might be mapped by fracture seismic methods, as described by Sicking and Malin (2019) [1], means that, for the first time, the location of the major flow channels at a particular site can be determined and specified in models. The self-similar nature of permeability and the standard deviation of ln permeability from its mean allows the flow-field to be modelled in a statistically realistic way at finer scales. The combination of specific flow channel modeling with statistically realistic modeling of the finer details could produce more realistic and useful models of subsurface fluid flow for reservoir development and management, contaminant remediation, geothermal energy extraction, and many other purposes.

**Author Contributions:** Conceptualization, P.E.M., P.C.L.; methodology, P.C.L. and C.C.B.; writing-original draft, P.E.M. and P.C.L.; writing-review and editing, L.M.C., C.C.B. All authors have read and agreed to the published version of the manuscript

**Funding:** No external funding supported the content or preparation of this paper. **Acknowledgments:** The authors are grateful to Peter Geiser for ongoing discussion of the flow-structure seismic imaging matters central to this paper. We also acknowledge very helpful suggestions from the reviewers.

**Conflicts of Interest:** The authors declare no conflicts of interest.
