4.3 Investigation of Onset of Turbulent Flow in Rock Samples

Bulk density, total porosity, effective porosity, constant head, and permeameter test data were published for 13 cubes of Key Largo Limestone from southern Florida (DiFrenna et al., 2007). This limestone is denser and has smaller pores than the Biscayne Aquifer rock specimen shown in Figure 16b. Seventy percent of the permeameter tests conducted on the Key Largo Limestone samples remained under laminar flow, even with extreme gradients imposed during the permeameter experiments. Kuniansky and others (2008) fit the data for flow measurements along three orthogonal axes of the Key Largo Limestone cube number 6, one of the samples which exhibited a nonlinear relation between hydraulic gradient and discharge (DiFrenna et al., 2007). This nonlinear relation is indicative of non-Darcian flow. Cube number 6 measures 0.2 m on an edge with bulk density of 1.38 g/cm³, total porosity of 0.49, effective porosity of 0.34, and a representative pore diameter of 0.01 m (DiFrenna et al., 2007). The range of laminar hydraulic conductivity, K, was 36 to 61 meters per day and Re_c for all axes was 1.44 (Kuniansky et al., 2008). Thus, this carbonate exhibits the onset of non-Darcian flow at small Re_c. The sample has an average pore diameter at the upper limit of typical porous media, most of the samples had smaller pores and thus behaved as porous media.

Cunningham and others (2009) published hydraulic conductivity values for samples of the Biscayne Aquifer in south Florida, which is predominantly layers of secondary macro porosity limestone created by a large macropore network resulting from biologic activity. These samples of aquifer rock have pore diameters generally greater than 10 mm with the largest diameter being 300 mm. Laboratory determination of hydraulic conductivity was not always possible. For four samples, K ranged from 4 to 3,000 m/day from smallest to largest macro porosity. The impermeable and 300 mm pore size samples were omitted from the study. Sukop and others (2013) wrote about non-Darcian flow for models of these samples using lattice-Boltzmann simulations and found that flow begins to be non-Darcian at Re > 0.1 and is most likely turbulent between Re of 1 to 10.

In fractured rock systems, where flow occurs predominantly within the fractures between surfaces of less permeable rock, Quinn and others (2011a, b) conducted numerous short-interval borehole hydraulic tests and determined non-Darcian flow occurred at Re_c values between 0.1 and 6 for a fractured dolostone, which was in agreement with critical Re_c from laboratory experiments of single fractures in unidirectional flow where Re_c ranges between 1 to 10 (Konzuk and Kueper, 2004; Nicholl et al., 1999; Zimmerman et al., 2004).

To summarize, for most porous media, turbulent flow cannot be induced even with imposing unnaturally large gradients. For karst aquifers with layers that do not have dissolution features but have interconnected macropores without dissolution conduits, turbulence is possible and occurs at small Reynolds numbers more typical of large-pore, granular aquifers. The onset of turbulence in large dissolution conduits that behave more like pipes, is discussed briefly in Section 4.4 of this book.