The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

/pdf/validity-of-cubic-law-for-fluid-flow-in-a-deformable-rock-2ynq4id3hq.pdf

VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

[1] Laboratory analysis of single-phase water and airflow through a real rock joint under increasing confining pressure was carried out in order to assess the linearity or otherwise of the Reynolds number (Re) versus pressure change relationship and to also test the applicability of Forchheimer's relation. These tests were conducted on a granitic specimen, 110 mm high and 55 mm in diameter, with a single natural fracture running the height of the specimen, which had a matrix permeability of approximately 10−18 m2. The single-phase Re versus pressure change relationship using water was confirmed as being quadratic in nature. This same relationship held for single-phase airflow but only for confining pressures below around 2.0 MPa. At confining pressures ≥3.0 MPa the relationship shifted in form to a cubic fit. In analyzing the application of Forchheimer's transmissivity relation to single-phase water flow it was concluded that the model was appropriate for all tested confining pressures (0.55–3.0 MPa). When assessing its applicability to single-phase airflow, the model was only accurate at low-confining pressures (<1.0 MPa). Further, it was found that an increase in confining pressure shifted the Forchheimer's transmissivity curve downward.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2006WR005457

Nonlinear single-phase flow in real rock joints

Applicability of the 'cubic law' for non-Darcian fracture flow

In the last decade, because of the shale gas revolution in the USA the fluid flow in shale nanopores have attracted great attention of scientists worldwide. In shale formation water and natural gas can co-exist within the narrow pores, leading to the possibility of two-phase flow. In this work, I designed the molecular simulation system, that include water and methane confined in slit-shape muscovite nanopore, to investigate the effect of the two-phase flow patterns on the fluids transport and on the pore structure. The results indicate that when the driving force, i.e., the pressure drop, increases above a pore-size dependent threshold the two-phase flow pattern is altered. As a result, the velocity of methane with respect to that of water changes. My results also illustrate the importance of the capillary force, due to the formation of water bridges across the clay pores, not only on the fluid flow, but also on the pore structure, in particular its width. When the water bridges are broken, perhaps because of fast fluid flow, the capillary force vanishes leading to significant pore expansion. Because muscovite is a model for illite, a clay mineral often found in shale rocks, these results advance our understanding regarding the mechanism of water and gas transport in tight shale gas formations.

/pdf/water-and-methane-in-shale-rocks-flow-pattern-effects-on-3j1r049cwz.pdf

Water and methane in shale rocks: Flow pattern effects on fluid transport and pore structure

Mechanical behaviour of wellbore materials saturated in brine water with different salinity levels

Characterization of two-phase flow in a single rock joint

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Characterization of two-phase flow in a single rock joint