2. Permeability of Soil and Seepage Force 31
2.3 Groundwater Movement and Darcy’s Law
2.3.1.1 Laminar flow
Due to the groundwater seepage, the flow lines formed by the water particles are parallel in all places. The soil presents at the bottom
cannot float upward; leaves or other lightweight objects that float on the water surface are retained on the surface and cannot be pulled downward. Through a certain space, water flows smoothly with uniform velocity and the flow velocity in the midst of water cross-section is higher, but is smaller on both sides. Based on the flow characteristics stated above, the flow is commonly referred to as laminar flow. The flow velocity, flow direction, water level, hydraulic pressure, and a few other motion indexes at any point in this seepage field don’t change over time, which is called steady flow movement.
2.3.1.2 Turbulent flow
Compared with laminar flow, in this kind of groundwater seepage, the flow lines cross each other. The flow presents twisted, mixed, and irregular movements and exists as hydraulic drops and whirlpools.
Based on the flow characteristics stated above, this flow is commonly referred to as turbulent flow. All the motion indexes at any point in this seepage field change over time, which is called unsteady flow movement.
2.3.2 Darcy’s law
The voids in a soil (sand, clay) are generally quite small. Although the real seepage of water through the small void spaces in a soil is irregular, the flow can be regarded as laminar flow, because the movement of water through the void spaces is very slow. Under the condition of laminar flow, the discharge velocity and energy loss obey the linear seepage relationship, which was obtained by Darcy in 1856.
In order to obtain a fundamental relation for the quantity of seepage through a soil mass under a given condition, the case shown in Fig. 2.2 is considered. The cross-sectional area of the soil is equal to A and the rate of seepage is Q. Darcy experimentally found that Q was proportional to i, that is
Q=k·A·i. (2.1)
Note that A is the cross-sectional area of the soil perpendicular to the direction of flow.
The hydraulic gradient ican be given by i= H1−H2
L = ΔH
L ,
where Lis the distance between the two piezometric tubes.
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Fig. 2.2. Permeability test device.
A further consideration of the velocity at which a drop of water moves as it flows through soil helps to understand fluid flow.
Equation (2.1) can be rewritten as Q
A =k·i=v.
Therefore, for sandy soils, Darcy published a linear relation between the discharge velocity and the hydraulic gradient
v=ki, (2.2)
where v is the discharge velocity (cm/s), which is not the actual velocity of seepage through soil, and the velocityvgiven by Eq. (2.2) is the discharge velocity calculated on the basis of the gross cross- sectional area; iis the hydraulic gradient, the water-level difference per unit length along the flow direction; k is the coefficient of permeability, cm/s, a measure of the resistance of the soil to flow of water.
Darcy’s law given by Eq. (2.2), v =ki, is valid for laminar flow through the void spaces. Several studies have been conducted to investigate the range over which Darcy’s law is valid, and Reynolds number was obtained as a result of an excellent summary of these
works. For flow through soils, Reynolds number Rn can be given by the relation
Rn= vDρ
μ , (2.3)
where v is the discharge velocity (cm/s),D is the average diameter of the soil particle (cm), ρ is the density of the fluid (g/cm3), μ is the coefficient of viscosity (g/(cm s)).
For laminar flow conditions in soils, experimental results showed that
Rn= vDρ μ ≤1.
With coarse sand, assume D = 0.45 m and k ≈ 100D2 = 100(0.045)2 = 0.203 cm/s.
Assumei= 1, then we getv=ki= 0.203 cm/s. Also, letρwater ≈ 1 g/cm3 and μ20◦C = (10−5)(981) g/(cm s). Hence, we obtain
Rn= (0.203)(0.045)(1)
(10−5)(981) = 0.931<1.
From the above calculations, we can conclude that, flow of water through all types of soil (sand, silt, and clay) is laminar and that Darcy’s law is valid. With coarse sand, gravels, and boulders, turbulent flow of water can be expected.
Darcy’s law as defined by Eq. (2.2) implies that the discharge velocity bears a linear relation with the hydraulic gradient for sand (Fig. 2.3(a)).
Fig. 2.3. Discharge velocity–hydraulic gradient relationship of soil. (a) Sand, (b) clay, and (c) gravel.
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Hansbo (1960) reported the results of four undisturbed natural clays (Fig. 2.3(b)). On the basis of his results, it is seen that
v=k(i−i0), i≥i0, and
v=kin, i < i0.
The value of n for the four Swedish clays was about 1.6. There are several studies, however, that refute the preceding conclusion.
For gravel or other coarse-grained soils, the discharge velocity bears a linear relation with the hydraulic gradient just when the hydraulic gradient is small; when the hydraulic gradient is quite large, turbulent flow of water can be expected. Discharge velocity and hydraulic gradient will not present a linear relationship (Fig. 2.3(c)).
It must be pointed out that the velocityvgiven by Eq. (2.3) is the discharge velocity calculated on the basis of the gross cross-sectional area. Since water can flow only through the interconnected pore spaces, the actual velocity of seepage through soil is v. Assuming the rate of seepage is Q, the cross-sectional area of the soil is equal to A. Hence, the actual cross-water area
A =nA.
According to the continuity of water, Q = vA = vA, then we have
v =v×A
A =vn=v e
1 +e, (2.4)
where eis the void ratio of the soil andnis the porosity of the soil.
2.4 Determination of Permeability Coefficient