# Steady-State Water Flow: Theory, D. L. Nofziger

Program Documentation

General Description Model Description Simplifications Glossary Bibliography and Contributors

**Steady-State Water Flow in Porous
Media**

The rate of water flow and the distribution of water potentials in a one-dimensional soil column under steady-state conditions can be obtained by solving the Darcy or Buckingham-Darcy equation. When soils are saturated and homogeneous, the calculation is easy. The problem becomes more difficult when the soil is not homogeneous or when it is unsaturated. This program uses numerical methods to evaluate the flow rate of water, the equivalent conductivity of the soil, and soil-water potentials in a one-dimensional flow system composed of one or two layers of soil. The flow rate, equivalent conductivity, and a graph of total potential, matric potential, gravitational potential, conductivity, driving force, or water content as a function of position along the soil column are then displayed.

**Model Description**

Darcy’s equation is often used to describe water movement in saturated soils. For one-dimensional water flow, it can be written as ( Hillel, 1982)

where q is the flux density of water passing through the soil, K is the saturated hydraulic conductivity, H is the total potential, and x is the position coordinate.

For unsaturated soils, this equation takes the modified form developed by Buckingham (1907)

where K(q) or K(h) is the unsaturated hydraulic conductivity function (Brooks and Corey, 1964; Gardner, 1958; vanGenuchten, 1980), q is the volumetric water content and h is the matric potential. The total potential is the sum of the matric potential and gravitational potential.

For uniform saturated soils we can write this equation as

where H_{1} and H_{2} are the total potentials at
the inlet and outlet of the soil system, respectively. The difference
in total potential divided by the length of the system is the driving
force causing water to flow. This equation is applicable to layered or
unsaturated soils if the hydraulic conductivity, K, in the equation is
regarded as the equivalent
conductivity of the entire soil system.

When soils are saturated and homogeneous, the flux density can be easily obtained if the conductivity of the soil is known and the total potential or matric potential at the ends of the soil are specified. However, the problem becomes more difficult when the soil is not homogeneous or when it is unsaturated.

This program considers one-dimensional flow in a soil. The soil is assumed to be either homogeneous or composed of two layers. In a homogeneous soil system, the conductivity will be uniform throughout if the matric potentials at both ends are zero or more since that means the soil is saturated. If one or more matric potentials are less than zero, the conductivity and water content can change with position. In a layered soil system, the conductivity will not be uniform throughout even if the soil is saturated.

The program allows a user to select a soil system made up of one or two layers from the drop-down "Preferences" menu. If a two-layer system is selected, the user can specify the length or thickness of each layer in the soil system, the saturated hydraulic conductivity of each layer, the total potential or matric potential at each end of the soil, and the orientation of the soil. The user can also define for each layer the water content and hydraulic conductivity as functions of matric potential that are needed for unsaturated flow. When an one-layer system is selected, the user only needs to specify the thickness and hydraulic parameters for the single layer along with the potentials at the ends of the soil system and its orientation.

The software then solves the equations above for the system and displays graphs of total potential, matric potential, gravitational potential, conductivity, driving force, or water content as a function of position along the soil column. In this way the user can see the impact of changes in matric potential and water content upon the flow rate and equivalent conductivity of the entire soil system.

**Assumptions and
Simplifications**

Simplifications and assumptions on soil media and flow conditions:

- This software assumes that flow is strictly one-dimensional.
- Flow-induced deformation in a soil medium affects the shape, size, and connectivity of soil pores, which largely determines the hydraulic properties of a soil system ( Buckingham, 1907). In this program, it is assumed that the soil system simulated in this program is rigid, i.e., no flow-induced deformation occurs.
- Flow through conduits is classified into laminar flow and turbulent flow dependent on the ratio of inertial to viscous forces exerted on the fluids( Bear, 1972). In laminar flow, the relationship between flow rate and gradient in total potential is linear, while in a turbulent flow the relationship is nonlinear. The proportional relationship of Darcy's law is valid in the range of laminar flow only. Hence, an implicit assumption in this program is that flow conditions inside soil pores are laminar.
- In fine-grained (especially clayey) soils, a minimum potential gradient called threshold gradient may exist. Below this threshold gradient there is very little flow. This non-Darcy behavior(Swartzendruber, 1962) may be due to the effect of stream potential which generates small countercurrents along the pore walls in a direction opposite that of the main flow (Bear, 1972). Another explanation for the non-Darcy behavior is the non-Newtonian liquid viscosity caused by clay-water interaction. In this program, non-Darcy behavior is ignored.

*Equivalent Conductivity*: The
equivalent conductivity of a non-homogeneous soil is equal to the flux
through that soil divided by the driving force or gradient in total
potential across that soil. It could be considered the equivalent or
average conductivity for the non-uniform soil.

*Flux Density*:
The flux density of water passing through a soil is the volume of water
passing through the soil per unit cross-sectional area (perpendicular
to the flow) per unit time. It has units of length per unit time such
as mm/sec, mm/hour, or inches/day. Many times flux and flux density are
used interchangeably.

*Gravitational
Potential*: The gravitational potential of water is the amount of
work required per unit quantity of water to move a very small amount of
water reversibly and isothermally from a pool of pure water at
atmospheric pressure at a reference level to another pool of pure water
at the elevation of interest. This is simply the amount of work
required to lift or lower the water from the reference level. (See also
Units of Potential).

*Hydraulic Conductivity*: The
hydraulic conductivity of a soil is a measure of the ease at which
water moves through the soil. It can be obtained experimentally by
measuring the flux density of water passing through a soil, the
difference in total potential and the length of the soil. The
conductivity is the proportionality constant which when multiplied by
the driving force (or gradient in total potential) causing water to
move gives the flux density of water. If the potential is defined in
terms of a unit weight of water, then the gradient in total head has no
dimensions and the conductivity has units of length per unit time just
as the flux density does.

*Matric
Potential*: The matric potential of water in a soil is the amount
of work required per unit quantity of water to move a very small amount
of water reversibly and isothermally from a pool of pure water at the
elevation of interest and at atmospheric pressure to the point of
interest in the soil. This is the amount of work required to move water
into a soil from outside of it. Since the elevations are the same,
gravity has no impact upon matric potential. Matric potential is
another term for pressure potential or pressure head. (See also Units of Potential)

*Total
Potential*: The total potential of water in a soil is the amount of
work required per unit quantity of water to move a very small amount of
water reversibly and isothermally from a pool of pure water at
atmospheric pressure and at a reference level to the point of interest
in the soil. This is the sum of the matric potential and the
gravitational potential. (See also Units of
Potential)

*Units of
Potential*: All definitions of potential refer to work per unit
quantity of water. The final units of potential depend upon the unit
quantity of water chosen. It is convenient to define potential per unit
weight of water. This means that all types of potential have units of
length. This form of potential is often called "head". So we commonly
talk of total head, gravitational head, and pressure head or matric
head.

*Volumetric
Water Content*: The volume of water in a soil divided by the total
volume of the soil (i.e. the sum of the volumes of solids and
pores).

**Bibliography**

Bear, J., 1972. Dynamics of fluids in
porous media. Dover Publications, Inc., 31 East 2^{nd} Street,
Mineola, NY 11501, pp. 119-194.

Brooks, R. H., and A. T. Corey, 1964. Hydraulic properties of porous media. Colorado State University, Fort Collins, Colorado, Hydrology Paper no. 3.

Buckingham, E., 1907. Studies on the movement of soil moisture. U.S. Dept. of Agr. Bur. Soils Bull. 38.

Gardner, W. R., 1958. Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci. 85:228-232.

Hillel D. 1982. Introduction to soil physics. Academic Press, Inc. 1250 Sixth Avenue, San Diego, CA 92101.

Swartzendruber D. 1962. Non-Darcy flow behavior in liquid-saturated porous media. J. Geophys. Res. 67:5205-5213.

van Genuchten, M. Th., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892-898.

**Contributors**

This program was designed by Dr. D. L. Nofziger and Dr. J. Wu, Department of Plant and Soil Sciences, Oklahoma State University, Stillwater, OK 74078.

Send email to david.nofziger@okstate.edu

Last Modified: January 16, 2008.