Darcy Law for Flow of Water in Soil | Water Flow | Soil Management

In this article we will discuss about the Darcy law for flow of water in soil.

Flow of Water in Horizontal Saturated Column:

If the soil is simply a bundle of straight and smooth tubes, each uniform in radius, the rate of water flow will be equal to the sum of the separate flow rates through the individual pores. Since, the soil pores are highly irregular, flow through the soil pores is limited by a number of constraints.

Experience shows that the flow rate Q, being the volume V flowing through the column per unit time, is directly proportional to the cross-sectional area and to the hydraulic head drop ΔH and inversely proportional to the length of the column L.

Q = V/t ∝ [(A x ΔH)/L]

Hydraulic head drop across the system is usually determined by measuring the head at inflow boundary (H_i) and at outflow boundary (H_o), relative to some reference level (Fig. 7.19). ΔH is the difference between the two heads.

∆H = H_i – H_o

Obviously, no flow occurs in the absence of hydraulic head difference (∆H = 0).

The head drop per unit distance in the direction of flow (∆H/L) is the hydraulic gradient, which is the driving force. The specific discharge rate Q/A (the volume of water flowing through a cross-sectional area A per time t) is called the flux density (simply flux) indicated by q. Thus, the flux is proportional to hydraulic gradient.

q = Q/A = [V/(A x t)] ∝ ΔH/L

The proportionality factor k is generally designated as hydraulic conductivity.

q = k x ∆H/L

This equation is known as Darcy law after Henry Darcy. Where flow is unsteady (flux changing with time) or soil non-uniform, the hydraulic head may not decrease linearly along the direction of flow. Where the hydraulic head gradient or the conductivity is variable, we must consider the localised gradient, flux and conductivity values rather than overall values for the soil system as a whole. A more exact and generalised expression of Darcy law is-

q = – k x ∆H

Stated verbally, this law indicates that the flow of a liquid through a porous medium is in the direction of and at a rate proportional to the driving force acting on the liquid (hydraulic gradient) and also proportional to the conductivity. The negative sign (–) indicate that the flow is in the direction of gradient or the direction of flow is towards decreasing head.

Flow of Water in Vertical Saturated Column:

Flow of water in a vertical saturated column (Fig. 7.20) indicate that the upper surface is under constant head of ponded water H₁ and the lower surface in constant level reservoir. Flow is from higher to lower reservoir through a column length L.

The hydraulic head gradient, which is the ratio of hydraulic head drop (between inflow and outflow) to the column length is necessary for calculating flux according to Darcy law.

The rate of downward flow of water in vertical column (Fig. 7.20) is greater than in horizontal column (Fig. 7.18) by the magnitude of hydraulic conductivity. If the ponding depth H_i, is negligible, flux is equal to the hydraulic conductivity.

In the case of upward flow in saturated vertical column (Fig. 7.21), the direction of flow is opposite to the direction of gravitational gradient and the hydraulic gradient becomes-

In a saturated soil of stable structure, as well as in a rigid, porous medium such as sandstone, the hydraulic conductivity is constant ranging from 10^-2 to 10^-3 cm s^-1 in a sandy soil and 10^-4 to 10^-7 cm s^-1 in a clayey soil. Hydraulic conductivity is greater if the soil is porous, fractured or aggregated than if it is tightly compacted and dense.

Gravelly or sandy soil with large pores has greater conductivity than a clayey soil with narrow pores, through the porosity of a clay is greater than that of a sandy soil.

In many soils, the hydraulic conductivity does not remain constant. Changes in the composition of exchangeable ion complex when water enters the soil can greatly change the hydraulic conductivity. In general, the conductivity decreases with decreasing concentrations of electrolytic solutes due to swelling and dispersion.

Detachment and migration of clay particles during prolonged flow may result in clogging of pores. Entrapped air may block pore passages. Hydraulic conductivity is not an exclusive property of the soil alone, since it depends on attributes of the soil and of the fluid together. Soil characteristics such as total porosity, distribution of pore sizes and tortuosity (pore geometry of the soil) affect the hydraulic conductivity.