# Exercise 10 Solution

##### Part 1

Temperature changes the viscosity and density of water. Thus, Reynolds did experiments at different temperatures.

##### Part 2

The table below shows the calculated Reynolds number for the different aquifer types and at different radial distances from the production wells from Exercise 9.

How to convert centistokes to square meters per second (cSt to m2/s):

ν m2/s = 1.0 × 10–6 × ν cSt

Hint regarding conversion of units for kinematic viscosity, consider how many square meters per second are in a centistoke: If, ν cSt = 1, then ν m2/s = 1.0 × 106 × 1 = 1.0 × 106 m2/s.

Note: a centistoke is a centimeter-gram-second (CGS) unit of kinematic viscosity. Square meter per second (m2/s) is a metric unit of kinematic viscosity. At 20o C, water has a kinematic viscosity of approximately 1 centiStoke. To convert “per second” to “per day”, remember 24×60×60 seconds in a day. So, 1 centiStoke is approximately 0.0864 m2/d.

Calculated Reynolds number for different aquifer types at different radial distances from production wells.

 Aquifer type Pumping Rate (m3/d) Thickness (m) Radial Distance in Meters 0.25 0.5 1 5 10 Reynold’s Number Alluvial, K=10 m/d, and average pore diameter 0.005 m 300 10 1.11 0.55 0.28 0.06 0.03 same 300 50 0.22 0.11 0.06 0.01 0.01 same 300 100 0.11 0.06 0.03 0.01 0.003 Point Bar gravel, K=100 m/d and average pore diameter 0.02 m 1000 10 14.74 7.37 3.68 0.74 0.37 same 1000 50 2.95 4.09 2.05 0.41 0.2 same 1000 100 4.09 2.05 1.02 0.2 0.1 Sandstone, K=1 m/d and average pore diameter 0.001 m 100 10 0.074 0.037 0.018 0.004 0.002 same 100 50 0.015 0.007 0.004 0.001 0.0004 same 100 100 0.007 0.004 0.002 0 0.0002

Shaded in blue are any estimated Reynolds numbers greater than one. A value >1 was chosen because when rock samples are tested in a lab, rather than glass spheres, the onset of divergence from Darcy’s law often occurs at Reynolds numbers >1. The sandstone never has an estimated Reynolds number greater that one, so flow is probably never non-Darcian in the sandstone aquifer. The alluvial aquifer may have non-Darcian flow very close to the well bore, but only for a small distance from the well where the velocity is highest. The largest hydraulic conductivity is associated with clean gravel in a large point bar deposit with a K of 100 m/day and this material may have non-Darcian flow further out into the formation from the well. This exercise reveals that it is possible for non-Darcian or even turbulent flow to occur near water supply wells in rock formations with a hydraulic conductivity >10 m/d. However, this non-Darcian flow will not extend far into the formation unless the hydraulic conductivity is 100 m/d or greater. These are estimated Reynolds numbers because for groundwater it is not easy to know the average pore diameter within a hydrogeologic unit. For granular aquifers, grain-size distribution via sieve analysis and total porosity estimates are commonly conducted. In sieve analysis, the notation, Dxx, refers to the size D, in mm, for which xx percent of the sample by weight passes a sieve mesh with an opening equal to D. The D10, sometimes called the effective grain size, is the grain diameter for which 10 percent of the sample (by weight) is finer and is sometimes used as an estimate of effective average pore diameter. Others have used the D50 size. In some cases, the height of the capillary fringe is used in calculations of effective pore diameter in soil physics and for intact rock samples measurements of fracture aperture are used or when samples are not available, aperture is estimated from photos of borehole walls are used to estimate the average effective pore diameter. The following materials discuss methods of estimating effective pore diameter: Nimmo, 2013; Glover and Walker, 2009; Revil and others, 2011; Fu and others, 2020.