Modern Techniques in Well Design - 2
Well Design
Critical Radius. Turbulence occurs in
the near zone of the well filter when inertial forces predominate over viscous forces.
The transition between completely laminar and completely turbulent is evidenced by the
gradual scatter of turbulence throughout all pores of the medium. Thus, the first
deviations from Darcy's law sorrespond to the beginning of appreciable eddy losses in
the larger pores. These localized regions of turbulence spread to the smaller pores as
velocity increases (Figure 2).
Figure 2
The following equations are based on Darcy's law and the Darcy -
Weisbach formula, and apply equally to uniform aquifers, filter packs (as used in the
model), and field conditions. Darcy's law shows that for laminar flow, the hydraulic
gradient dh/dr over a column of sand with a characteristic grain diameter
d, kinematic viscosity v, Reynolds number R, and intrinsic
permeability k may be expressed as:
dh/dr = a2v2R/gkd (1)
Similarly, for turbulent flow the head loss is proportional to the
square of the velocity (Darcy- Weisbach formula) or in terms of Reynolds number:
dh/dr = a2v2R2/gkd (2)
In which R = Reynolds number = vd/ v; v = velocity;
k = intrinsic permeability; d = characteristic grain diameter (assumed =
50 percent passing size); v = kinematic viscosity of water (1.233 ×
10-5 sq. ft/s); a1 , a2; = constants
(a1 = 1 for laminar flow, a2 = 1/R ); and g = gravitational
constant (32.2 fps2 [9.8 m/s2; ]).
The characteristic length in the standard definition of Reynolds
number has been replaced by a characteristic grain diameter, assumed equal to the 50
percent passing grain size. This substitution was made because the microscopic
passageways occupied by water in porous media are difficult to measure. At the point of
transition from laminar to turbulent flow (defined as the critical point and denoted by
the subscript c), the hydraulic gradients are equal and Eqs 1 and 2 can be equated,
yielding:
(dh/dr)c = v2/gkd)Rc = a2(v2/gkd)RV2 (3)
and
Rc = 1/a2
The constant a2 is equal to the reciprocal of the
critical Reynolds number (Rc) and is primarily a function of grain
shape, packing, and distribution. To test this relationship as well as determine the
critical Reynolds number, a series of experiments was performed using the well- aquifer
model.
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