MECAG 309HYDRAULIC CONDUCTIVITY1
Water Flow Through Soil (q = -K i)
- Flow in Capillary Tubes
2. Flow in Soil; Macroscopic flow
3. Using Darcy's Equation
4. Unsaturated flow
INTRODUCTION
The rate of water movement through soil is of considerable importance. The entry of water into soil, the movement of water to plant roots, the flow of water to drains and wells, and the evaporation of water from the soil surface are but a few of the obvious situations in which the rate of movement plays an important role. An important soil property involved in the behaviour of soil water flow is the conductivity of the soil to water. Qualitatively, hydraulicconductivity is the ability of the soil to transmit water (Klute, 1965). Knowledge of the hydraulic conductivity for soil materials provides the ability to properly design water control structures, earthen storage facilities, and provide improved flood and runoff forecasting. This section will provide a very general overview of flow process in porous materials.
1. Flow in Capillary Tubes (microscopic flow)top
Soil pores are capillary sized and thus the principles of water flow through them may be approximated by flow in capillary tubes of constant diameter. Flow results from a differential pressure placed in the tube. The pressure gradient (steepness of slope) is determined the by change in pressure over the distance of flow, (P1-P2)/L. How fast the water flows in the tube, in response to the pressure gradient, is determined by the interaction of the water molecules with the tube sides (adhesion) and with other water molecules (cohesion). This is determined by the size and shape of the tube ( r4/8) and by the viscosity of the water ();
Fig. 1. Representation of flow through a capillary tube.
The above equation provides the volume of water flowing per unit time (m3/s). If we divide through the by area of the tube (A, m2) then the equation provides flux (m3 m-2 s-1, or just m/s) If for simplicity it can be assumed that the viscosity of water stays constant (at 20ºC viscosity is 0.001 N s /m2), the gradient is 1 Pa/m (note a Pascal is N/m2), and Area ( r2) is divided into this equation then the following 'working' equation is achieved:
q (V A-1 t-1, m/s) = 125 r2
Thus a tube 0.1 mm in radius (1 x 10-4 m) will have a flux of 1.25 x 10-6 m/s, a tube 1 mm in diameter (1.25 x 10-4 m/s), etc.
2. Flow in Soil; Macroscopic flowtop
The flow of water through soil was first quantified by Henri Darcy (1856), who reported on the filtration of water flowing through a saturated sand bed for an improved supply for the city of Dijon (Marshall and Holmes, 1979). Water moves through soil in response to a pressure gradient. The volume of water (V) that moves through a given cross-sectional area of soil (A) in a specified period of time (t) is proportional to: the driving force (potential difference, dHt) and inversely proportional to the distance over which the driving force is exerting itself (dz);
V A-1 t-1 dHt/dz
The proportionality factor, , represents the ability of the soil to transmit water and can be termed hydraulic conductivity (K):
q = - K i
Fig. 2. For saturated soil the flux (q) is proportional to the gradient (i). The slope of q plotted against i is K, hydraulic conductivity.
Where q is the flux density (V A-1 t-1). And i is hydraulic gradient (dHt/dz). This equation describes the basic physical relationship governing the flow of water under certain conditions). The flow of water under certain conditions, known as Darcy's Lawstates that the flow of a liquid through a porous medium is in the direction of force, and at a rate proportional to the driving force acting on the liquid (i.e., the hydraulic gradient) and the proportionality factor is the property of the conducting medium to transmit the liquid or the hydraulic conductivity
Table 1. Saturated hydraulic conductivity values of some general materials
m/s / mm/hr / cm/d / mm/yr / Material1.0E-03 / 3.6E+03 / 8.6E+03 / 3.2E+07 / Coarse sands
1.0E-04 / 360 / 864 / 3.2E+06 / Sands
1.0E-05 / 36 / 86.4 / 3.2E+05 / Most soils
1.0E-06 / 3.6 / 8.64 / 3.2E+04 / Clay/weathered till
1.0E-07 / 0.36 / 0.864 / 3.2E+03 / Heavy Clays
1.0E-08 / 0.036 / 0.0864 / 315.3 / Solonetzic soils
1.0E-09 / 0.0036 / 0.00864 / 31.5 / Unweathered till to
1.0E-10 / 3.6E-04 / 8.6E-04 / 3.2 / - compacted clay barriers
Fig. 3 Saturated hydraulic conductivities of North American agricultural soils
Compaction from animals and equipment can decrease saturated hydraulic conductivities by more than one order of magnitude. Solonetzic B horizons can be less than 1 E-8 m/s.
Fig. 4. Variation of porosity and saturated hydraulic conductivity with depth for Prairie materials.
Hydraulic conductivity, permeability, and fluidity
The hydraulic conductivity is the proportionality factor between the flux and the hydraulic gradient or the ratio of the flux to the hydraulic gradient or the slope of the flux vs the hydraulic gradient.
q = - Ki,
K = - q/i
The hydraulic conductivity, K, is a function of the properties of the medium (the soil) and the fluid (water). Soil properties that affect K are referred to as intrinsic permeability (k), whereas the fluid properties is the fluidity (liquid or gas f).
K = kf
When K is in units of m/s (LT-1), k is expressed in m2 (L2) and f in 1/(m s) or (L-1 T-1).
Fluidity is inversely proportional to viscosity (): f = g/
where is density of the fluid, g is acceleration due to gravity. Hence
K = k g/, or k = K /(g)
In an ordinary liquid the density is nearly constant (its variation due to temperature and solute concentration are minimal), and changes in fluidity are likely to result primarily form changes in viscosity (ie for water between 25 and 5°C the viscosity doubles). However for gas systems changes in density with pressure can be considerable.
The use of the term permeability can be confusing as it is often used synonymously with hydraulic conductivity. It is also used in a loosely qualitative sense to describe the readiness by which a medium transmits water or other fluids and gases. Consequently in the strict sense permeability should be used with a qualifier such as intrinsic or specific.
Thus permeability is ability of the media to transmit any fluid or gas as its properties is independent of what is being transmitted (as long as the fluid does not rearrange the pore structure). This occurs in a completely stable porous body (ie sand stone). In the same way the fluidility of a fluid or gas is irrespective of the medium.
Using Darcy's Equation top
As the property hydraulic conductivity can not be directly measured it must be determined by measuring flux and hydraulic gradient:
K = - q/i = (V dz)/ (At dHt)
EXAMPLEGIVEN: the soil column at the right. The soil column has an area of 80 cm2, has a constant head of 8 cm and after 7 hours 450 cm3 of water has been collected in the pan under the soil sample. The reference point is at point B.
FIND: the saturated hydraulic conductivity.
At = Ap + Az = 8 cm + 12 cm = 20 cm
Bt = Bp + Bz = 0 cm + 0 cm = 0 cm
= -1.36 x 10-4 cm/s /
Unsaturated Flowtop
Flow in most agricultural soils is unsaturated. At a microscopic scale flow is dependent upon the diameter of the capillary tube. In the previous chapter where soil water availability was discussed the soil moisture retention curve may be likened to bundles of capillary tubes of various sizes. As soils become unsaturated the largest pores lose their water first. Using the analogy of capillary tubes the suction that a certain capillary tube desaturates may be represented by the following equation:
where h is the suction (in m height of water) required to empty a pore of radius r, with a liquid of density l, with a surface tension of , and a liquid-air contact angle of cos . If is the surface tension of water is set at 0.073 N m-1 (20°C), the contact angle can be assumed to be zero for water against nonhydrophobic materials and thus cos . Becomes 1.0; and with gravity at 9.8 m s-2 and density of water at 1000 kg m-3 the equation reduces to:
h = 1.5 E-5/r orr = 1.5 E-5 / h; where r and h are in units of meters.
Consider the soil cross-section in the below figure and applying the equations of flow and retention for capillary tubes.
Fig. 5. Soil thin section obtained of an agricultural soil at a depth of 10 cm.
If 1% of the total pore space was occupied by pores of 0.5 mm radius, 10% pores at field capacity (0.005 mm) and the remainder of the pores that empty at 2.9 bars suction the total flow at saturation would be dominanted by the 1 % of large pores. The soil at field capacity will have a flux rate (and thus hydraulic conductivity 5 orders of magnitude smaller than at saturation. For this reason sandy soils, which are primarily composed of just large pores conduct very little water and result in drought conditions at the slightest desaturation.
Table 2. Estimation of flux (q) and flow rate (Q) using capillary rise and flow equations
Radius / suction / q / A / Qmm / bar / cm/d / % / cm3/d
0.5 / 0.003 / 270 / 1 / 2.7
5E-3 / 0.29 / 0.027 / 10 / 0.0027
5E-4 / 2.9 / 2.7e-4 / 89 / 2.4e-4
A is percentage of total pore area occupied by pores of set radius.
REFERENCES
Hillel, D. 1982. Introduction to Soil Physics. Academic Press. New York London, pp 90-107.
Klute, A. 1965. Laboratory measurement of hydraulic conductivity of saturated soil. In: C.A. Black (ed.) Methods of Soil Analysis, Part I: pp.210-221.
Marshall, T.J. and J.W. Holmes. 1979. Soil Physics. Cambridge University Press. pp. 81-99.