Previous newsletters have discussed equations that represent saturated groundwater flow, such as Darcy's Law.  However, there has been little mention of the equally important unsaturated flow.  Unsaturated flow occurs when not all of the pore spaces in the media are filled with water; there is another media, typically air, that is also in the pore spaces.  This effects many flow properties, such as the hydraulic conductivity of the medium.  The following will discuss some of the differences between the hydraulic conductivity of the saturated and unsaturated zones.

How does hydraulic conductivity change under unsaturated conditions?

How is Darcy's Law effected by this?

What is the relative permeability term?

How does hydraulic conductivity change under unsaturated conditions?

In general, the hydraulic conductivity of a medium will decrease rapidly with decreasing water content; the soil moisture cannot freely flow in the pores that contain air, the moisture typically flows along the particle surfaces of those pores. Under saturated conditions the hydraulic conductivity remains the same.  Once the medium is no longer fully saturated the hydraulic conductivity begins to decrease, and continues to decrease until no moisture is left in the media and the hydraulic conductivity is essentially zero. 

How is Darcy's Law effected by this?

Conventional Darcy's Law, as given in a previous newsletter, uses a constant term for hydraulic conductivity.  Obviously, this will not work under unsaturated conditions as the conductivity will change with moisture content.  The extension of Darcy's law for unsaturated flow therefore assumes that hydraulic conductivity is a function of the volumetric water content (creating a non-linear equation); this is done using the relative permeability term.

What is the relative permeability term?

The relative permeability term varies from zero to 1 (in field conditions), and is used to scale the hydraulic conductivity from it's fully-saturated state (relative permability value of 1) to the completely unsaturated state.  The equation for relative permeability is not universal, as it is empirically calculated.  One of the more common derivations is by Brooks and Corey, 1966:

The van Genuchten parameter n can be found for various soil types, and typically ranges from 1-2 for fine textured media, and up to 4 for coarse.  A list of some of the values is given below:

Texture Class	n
Clay	1.25
Loam	1.47
Sand	3.17
Silt	1.67

From USDA Rosetta Model list of class average hydraulic parameters (http://www.ars.usda.gov/Services/docs.htm?docid=8955)

A calculator for Brooks and Corey's relative permeability term is given below.

Brooks and Corey Relative Permeability Calculator

Ө_m:   

Ө_r: 

porosity: 

n: 

k_r:   atm m³/mol

Example Calculation

A fine textured soil has a saturation of 0.25, a residual saturation of 0.1 and a porosity of 0.3.  Using a van Genuchten n value of 1.2, what is the residual permeability?

Calculate Ө_m-Ө_r: 0.25-0.1 = 0.15

Calculate Ө_T-Ө_r: 0.3-0.1 = 0.2

Relative permeability = 0.75^1.2 = 0.71

There are many software products available for determining unsaturated flow.  One of these programs is the Visual MODFLOW addition, MODFLOW-SURFACT.

 

References

 

Tindall, J.A. and Kunkel, J.R. (1999). Unsaturated Zone Hydrology for Scientists and Engineers. Published by Prentice-Hall Inc., New Jersey.

Warrick, A.W. (2003). Soil Water Dynamics. Published by Oxford University Press, New York.

 

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Visual MODFLOW with MODFLOW-SURFACT

Visual MODFLOW is a widely used groundwater modelling program, with extensive capabilities.  Some of these capabilities are not available through the traditional MODFLOW environment, but have been made available through add-on programs.  Some add-ons include MIKE 11, a surface water modeling package, WinPEST, a parameter estimation package, and MODFLOW-SURFACT, a variably saturated groundwater flow package.  It is this program, MODFLOW-SURFACT, that will be the topic of this month's newsletter.

MODFLOW-SURFACT is a robust 3D finite-difference flow and transport program that includes several improvements over the public domain version of MODFLOW such as the ability to handle complex variably-saturated flow and transport.  Other capabilities include:

	Unsaturated vapor flow
	Adaptive time-stepping
	Solver incorporates Newton-Raphson linearization
	Multi-phase, multi-component solute transport
	Advective-dispersive transport with retardation
	Biochemical degradation
	many more!

Examples of when you would use MODFLOW-SURFACT include scenarios where you have multiple water tables or perched aquifers, steep flow gradients, infiltration through the vadose zone, and vapor flow through the unsaturated zone.

For more info click here:  Visual MODFLOW
For demo download click here: Demo Download
To purchase click here: Order Online

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