As we know, in the Forchheimer equation -▽P=μQ/K+βρQ2 (μ is the viscosity, K is the intrinsic permeability, ρ is the density), the coefficient β depends on the properties of the porous/fractured medium only. For porous media, there are many equations empirical or theoretical to determinate β, such as 1.75(1-n)/g/n3/D [Ergun,1952] (n is the porosity, D is the particle diameter, g is acceleration of gravity). But, what about the β for rough-walled fracture? Is there any equation to determinate β according to the fracture geometries？
Dear Dr. Chen,
Thanks for your concern about this question, and I have read your paper about how to determine β in coalbed methane production. Using empirical correlatoin is also a good way to determine β and I will have a try. In addition, my research work need to predict the β depending on the fracture geometries, such as the aperture, roughness, and contact area ratio, which is similar to the euqation of porous media. Is it possible to develop a way to determine β in such a way?
Thank you very much for the explanation!
Best regards!
Happy Mid-Autumn Festival
To evaluate Beta as a function of fracture aperture I would suggest to look at Reynolds Number and friction factors in parallel plates, there are some analytical solutions out there .
And for aperture and rugosity perhaps:
AN EMPIRICAL CORRELATION FOR ISOTHERMAL PARALLEL PLATE CHANNEL COMPLETELY FILLED WITH POROUS MEDIA AU - Hamdan Mohammad O JN - Thermal Science PY - 2013 VL - 17 IS - 4 SP - 1061 EP - 1070 PT - Article AB - This study reports a simple empirical correlation for friction factor and Nusselt number for laminar, steady state, hydraulically and thermally fully developed flow in isothermal parallel plate channel completely filled with porous media. The empirical correlations are developed to relate friction factor and Nusselt number to Darcy and Forchheimer coefficient.
Analytical investigation of forced convection from a flat plate enhanced by a porous substrate
Dr. A. V. Kuznetsov, Acta Mechanica
1999, Volume 137, Issue 3-4, pp 211-223
Bench scale laboratory tests to analyze non-linear flow in fractured media
C. Cherubini , C. I. Giasi, and N. Pastore.
Hydrol. Earth Syst. Sci., 16, 2511–2522, 2012
www.hydrol-earth-syst-sci.net/16/2511/2012/
doi:10.5194/hess-16-2511-2012
Dear Dr. Grattoni,
Thank you very much, your answer is very helpful. And I have got some inspiration from your answer.
Best regards!
Hi,
I suggest to read my paper "A new geometrical model for non-linear fluid flow through rough fractures" in Journal of Hydrology
Volume 389, Issues 1–2, 28 July 2010, Pages 18–30
In this paper, we proposed an equation to determined the coefficient you are locking them.
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