Knudsen-Like Scaling May Be Inappropriate for Gas Shales

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The author writes that the generally accepted Knudsen diffusion in shales is based on a mistranslation of the flow physics and may give theoretically unsound predictions of the increased permeability of shales to gas flow. This increase of permeability comes from the micropores, fine-scale microfractures, and cracks. The nanopores in shales provide gas storage by sorption and capillary condensation of heavier gas components. In the smallest nanopores, even methane molecules are increasingly ordered and resemble liquid more than gas. These nanopores feed the macroscopic flow paths in ways that are not captured well by generally accepted equations.


For gas pressures below 1 bar, gas permeability can exceed that of liquid substantially. Size distribution of a single pore is a distribution of radii of the largest spheres that can be fitted at each point along this pore. “Pore size” or “pore-body radius” is the radius of maximum sphere that can be inscribed into a pore, while “pore throat” refers to the radius of a minimum inscribed sphere common to two adjacent pores. In slit-like pores, pore throats and bodies are the same and pore widths are often reported to account for gas sorption. Pore sizes—whatever this term means to different authors—in the crushed samples of mudrocks are often inferred from small-angle and ultrasmall-angle neutron scattering, multistage desorption measurements, and molecular or statistical physics calculations; these sizes are not directly measured. A specific definition of pore size is provided in the complete paper.

The nanopores and micropores in sedimentary, compacted silicious and calcarious mudrocks (shales) are connected but have very low permeability. This low permeability results from the small cross sections of pore throats and the scale-dependent connectivity that ranges from strong at nanoscale to increasingly sparse at micro and higher scales. Methane in shales is produced through the highways of loosely connected micropores and multiscale cracks, both man-made and natural, into which a background continuum of the tiny nanopores feeds gas. This background continuum with the ordered, densely packed methane molecules can masquerade at times as multilayer methane adsorption.

Nanopore densities within grains of organic matter can be high. Grains containing hundreds of nanopores are common. The fractured gas-bearing mudrock formations (shales) are essentially multiscale and multiphysics systems, and their behavior is complex. Yet researchers frequently attempt to replace shale complexity with a set of slim cylindrical capillaries that flow methane with slip at the capillary walls. The majority of the complete paper describes a 180-year-long path of classical physics that has led to the models of Knudsen diffusion of gases in capillaries and the Knudsen flow regimes in the high-Reynolds-number flows of rarified gases, later transplanted to petroleum literature.

Poiseuille Flow

Jean Léonard Marie Poiseuille was interested in flow of human blood in narrow tubes. In 1838, he experimentally derived, and in 1840 and 1846 formulated and published, Poiseuille’s Law, now commonly known as the Hagen-Poiseuille equation. This law applies to laminar flow of liquids through tubes of uniform cross section, such as flow of water in slim glass capillaries or blood flow in capillaries and veins.

What distinguishes a gas from liquids and solids is the vast separation of the individual gas molecules. This separation is measured by the mean free path (\(\lambda\)) of gas molecules and usually makes a colorless gas invisible to the human observer. Therefore, the analogy of a large Knudsen number, devised originally for a very long \(\lambda\) and a large characteristic length of the solid, R, but such that the ratio \(\lambda\)/R is large, breaks down when both \(\lambda\) and R are close to the size of gas molecules but their ratio is still large.

All similar theories are, in fact, intermediate asymptotics and break down near the physical limitations on the sizes of important system parts. This is the reason for the author’s doubt about the applicability to mud rocks of the Knudsen-like scalings developed for the rarified gases in large tubes. The dense supercritical fluid in shale pores is more akin to a liquid than to a rarified gas.

Slip Flow

In small-scale low-velocity flow of gases, the failure of the standard Navier-Stokes description can be quantified by the Knudsen number, Kn=\(\lambda\)ℓ, where \(\lambda\) is the molecular mean free path calculated at a pressure that characterizes the flow and ℓ is the characteristic hydrodynamic length scale, here the capillary tube radius. On the basis of a discussion of slip velocity in the complete paper, the author writes that the widely accepted classification developed by Farzam Javadpour in 2009 is inconsistent with the flow physics.

The author argues that, if a cylindrical capillary analogy for gas flow in mud rocks is insisted upon, then the slip correction is either mostly irrelevant for larger pores or inappropriate for smaller pores. Because the increase of effective permeability to gas flow in mud rocks is real, one must invoke other pore geometries and admit that many, perhaps thousands, of very small nanopores packed with liquid-like dense gas connect to micropores.


To model gas flow in a mudrock system at pore scale, one needs a fairly complete description of geometry and connectivity of the flow network. To use the angular cylindrical capillary analogy for each pore body, a representative shape factor (ratio of pore cross-sectional area to the square of perimeter), inscribed circle radius, length, and coordination number must be known. The same parameters must also be known for each pore throat connecting this pore body to other pores. It is well known that the slit-like, angular capillaries can have hydraulic conductances that are much smaller than those of circular cylinders. A multiscale topological description of volumetric rock images with a hierarchy of inscribed spheres allows one to perform direct lattice Boltzmann calculations of fluid flow.

Mudrocks are essentially multiscale, have complex pore shapes, and currently feature an incomplete pore-level description. Even pore-size determinations are ill-defined, inconsistent, and often incomplete.

The author concludes that, in the smallest nanopores, flowing gas would slip if it did not behave more like a liquid. In the larger nanopores, at high reservoir pressure, the effect of slip is limited and likely smaller than uncertainty in the determination of absolute permeability and the permeability’s dependence on pore pressure. Therefore, amending the macroscopic Darcy equation with an ad hoc “Knudsen diffusion” term seems to lack predictive capability and is likely unwarranted. To account for permeability increase in gas flow, inclusion of gas transport through the porous walls of micro- and mesoscale capillaries might be useful.

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 187068, “Knudsen-Like Scaling May Be Inappropriate for Gas Shales,” by Tadeus W. Patzek, SPE, King Abdullah University of Science and Technology, prepared for the 2017 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 9–11 October. The paper has not been peer reviewed.

Knudsen-Like Scaling May Be Inappropriate for Gas Shales

01 September 2018

Volume: 70 | Issue: 9


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