Experimental observations show that flow through a fracture decreases more rapidly than the cube of the mean aperture (COOK, 1992). In order to provide a possible explanation of these experimental findings, we study creeping flow through a fracture of varying aperture that is symmetric about its midplane, using the power series of the stream function obtained by VAN DYKE (1987) for low Reynolds numbers. For the case of sinusoidally-varying walls, a simple expression relating the effective hydraulic aperture of the channel to the mean aperture and to the amplitude and wavelength of the sinusoidal wall profiles is obtained. Comparison is made to previous studies (KITANIDIS and DYKAAR, 1997) and to finite element calculations, and good agreement is obtained. The effect of fracture closure is then modelled as a decrease of the mean aperture without a change in the roughness. A power law relationship can be obtained between the flowrate and the mean aperture, with an exponent as high as 10, thus providing a potential mechanistic explanation of the experimental findings of PYRAK-NOLTE et al. (1987). Figure 3 Comparison of the results obtained with Fluidity (r) with the perturbation solution (-) and the Reynolds lubrication equation (---). The symbols correspond to the stagnant limit (BROWN et al.

Creeping flow of a Newtonian fluid through tubes of varying radius is studied. Using an asymptotic series solution for low Reynolds number flow, velocity profiles and streamlines are obtained for constricted tubes, for various values of constriction wavelength and amplitude. A closed-form expression is derived to estimate the pressure drop through this type of tube. The results obtained with this new expression are compared to data from previous experimental and numerical studies for sinusoidally constricted tubes. Good agreement is found in the creeping flow regime for the pressure drop versus flow rate relationship. Our method offers an improvement over the integrated form of the Hagen–Poiseuille equation (i.e., lubrication approximation), which does not account for the wavelength of the constrictions.

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Creeping flow through a sudden contraction/expansion in an axisymmetric pipe is studied. Sampson’s solution for flow through a circular orifice in an infinite wall is used to derive an approximation for the excess pressure drop due to a sudden contraction/expansion in a pipe with a finite expansion ratio. The accuracy of this approximation obtained is verified by comparing its results to finite-element simulations and other previous numerical studies. The result can also be extended to a thin annular obstacle in a circular pipe. The “equivalent length” corresponding to the excess pressure drop is found to be barely half the radius of the smaller tube.

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