The Engineer’s Continuum Mechanics versus Force Potential: an excursion into the physics of subsurface flow

K. Udo Weyer

In the proceedings of: GeoMontréal 2013: 66th Canadian Geotechnical Conference; 11th joint with IAH-CNC

Session: General Hydrogeology II

ABSTRACT: Within engineering practice, the calculation of subsurface flow is dominated by the mathematical physics of continuum mechanics. Continuum mechanics was born in the 19th century in an successful attempt to solve practical engineering problems. To that end were put in place quite a number of simplifications in geometry and the properties of water and other fluids, as well as simplifications of Darcy™s equation, in order to find reasonable answers to practical problems by making use of analytical equations. The proof of the correctness of the approach and its usefulness was in the practicability of results obtained. In the 1930s, a diametrically-opposed duality developed in the theoretical derivation of the laws of subsurface fluid flow between Muskat™s (1937) velocity potential (engineering hydraulics) and Hubbert™s (1940) force potential. The conflict between these authors lasted a lifetime. In the end Hubbert stated on one occasion that Muskat formulates a refined mathematics but does not know what it means in physical terms. In this author™s opinion that can still be said about the application of continuum mechanics by engineers to date. To date, engineering hydraulics is best represented by Bear (1972) and de Marsily (1986). In their well-known textbooks, both authors refer to Hubbert™s work as the proper way to deal with the physics of compressible fluids, as for example water. They then ignore, however, their own insights (de Marsily states so explicitly, Bear does not) and proceed to deal with water as an incompressible fluid. At places both authors assume the pressure gradients to be the main driving force for flow of fluids in the subsurface. That is not, however, the case. Instead the pressure potential forces are caused by compression initiated by unused gravitational energy not required to overcome the resistance to downward flow in penetrated rocks. In general the vectorial forces within gravitationally-driven flow systems are ignored when using engineering hydraulics. Scheidegger (1974, p. 79) states, however, verbatim and unequivocally: fiIt is thus a force potential and not a velocity potential which governs flow through porous mediafl (emphasis added). This presentation will outline the proper forces for groundwater flow and their calculations based on Hubbert™s force potential and additional physical insights. REFERENCES Bear, J. 1972. Dynamics of Fluids in Porous Media. American Elsevier Publishing Company, Inc., New York, NY, USA. de Marsily, G. 1986. Quantitative Hydrogeology: Groundwater Hydrology for Engineers. Academic Press, San Diego, California, USA. Hubbert, M.K. 1940. The theory of groundwater motion. Journal of Geology 48(8): 785-944. Muskat, Morris, 1937. The flow of homogeneous fluids through porous media. McGraw-Hill Book Company Inc., New York, New York, USA Scheidegger. A.E., 1974. The physics of flow through permeable media. Third Edition. University of Toronto Press, Toronto, Ontario, Canada

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