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

### **Peter E. Malin 1,\*, Peter C. Leary 2, Lawrence M. Cathles 3 and Christopher C. Barton 4**


Received: 11 November 2019; Accepted: 24 January 2020; Published: 28 January 2020

**Abstract:** Using Fracture Seismic methods to map fluid-conducting fracture zones makes it important to understand fracture connectivity over distances greater 10–20 m in the Earth's upper crust. The principles required for this understanding are developed here from the observations that (1) the spatial variations in crustal porosity are commonly associated with spatial variations in the magnitude of the natural logarithm of crustal permeability, and (2) many parameters, including permeability have a scale-invariant power law distribution in the crust. The first observation means that crustal permeability has a lognormal distribution that can be described as κ ≈ κ0 exp(α(ϕ − ϕ0)), where α is the ratio of the standard deviation of ln permeability from its mean to the standard deviation of porosity from its mean. The scale invariance of permeability indicates that αφ*o* = 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.

**Keywords:** crustal well-core poroperm; crustal fluid flow; crustal flow channeling; critical state physics; well-log spectral scaling; crustal power law scaling; lognormal; pink noise; crustal fracture seismics; crustal fracture imaging
