DARCY’S LAW
Darcy (1856) proposed that average fl ow velocity through soils is proportional to the gradient of the total head. The fl ow in any direction, j, is
where v is the average fl ow velocity, k is a coeffi cient of proportionality called the hydraulic conductivity (sometimes called the coeffi cient of permeability), and dH is the change in total head over a distance dx.
The unit of measurement for k is length/time, that is, cm/s. With reference to Figure 6.2, Darcy’s law becomes
where i 5 DH/L is the hydraulic gradient. Darcy’s law is valid for all soils if the fl ow is laminar.
The average velocity, v, calculated from Equation (6.7) is for the cross-sectional area normal to the direction of fl ow. Flow through soils, however, occurs only through the interconnected voids. The velocity through the void spaces is called seepage velocity (vs) and is obtained by dividing the average velocity by the porosity of the soil:
The volume rate of fl ow, qj, or, simply, fl ow rate is the product of the average velocity and the cross-sectional area:
The unit of measurement for qj is m3/s or cm3/s. The conservation of fl ow (law of continuity) stipulates that the volume rate of infl ow (qj)in into a soil element must equal the volume rate of outfl ow, (qj)out, or, simply, infl ow must equal outfl ow: (qj)in 5 (qj)out.
The hydraulic conductivity depends on;
1. Soil type: Coarse-grained soils have higher hydraulic conductivities than fi ne-grained soils. The water in the double layer in fi ne-grained soils signifi cantly reduces the seepage pore space.
2. Particle size: Hydraulic conductivity depends on D2 50 (or D21 0) for coarse-grained soils.
3. Pore fl uid properties, particularly viscosity: k1 : k2 < m2 : m1, where m is dynamic viscosity (dynamic viscosity of water is 1.12 3 1023 N.s/m2 at 15.68C) and the subscripts 1 and 2 denote two types of pore fl uids in a given soil.
4. Void ratio: k1 : k2 < e21 : e22 , where subscripts 1 and 2 denote two types of soil fabric for coarse-grained soils. This ratio is useful in comparing the hydraulic conductivities of similar soils with different void ratios. However, two soils with the same void ratio can have different hydraulic conductivities.
5. Pore size: The greater the interconnected pore space, the higher the hydraulic conductivity. Large pores do not indicate high porosity. The fl ow of water through soils is related to the square of the pore size, and not the total pore volume.
6. Homogeneity, layering, and fi ssuring: Water tends to seep quickly through loose layers, through fi ssures, and along the interface of layered soils. Catastrophic failures can occur from such seepage.
7. Entrapped gases: Entrapped gases tend to reduce the hydraulic conductivity. It is often very diffi cult to get gas-free soils. Even soils that are under groundwater level and are assumed to be saturated may still have some entrapped gases.
8. Validity of Darcy’s law: Darcy’s law is valid only for laminar fl ow (Reynolds number less than 2100).Fancher et al. (1933) gave the following criterion for the applicability of Darcy’s law for hydraulic conductivity determination:
where v is velocity, Ds is the diameter of a sphere of equivalent volume to the average soil particles, m is dynamic viscosity of water (1.12 3 1023 N.s/m2 at 15.68C), and g is the acceleration due to gravity. Typical ranges of kz for various soil types