STOMP

Directional Aqueous Relative Permeability Options (GT)

Directional/Anisotropic Relative Permeability Options

Constant
The relative permeability is constant regardless of saturation. 
Burdine

The Burdine (1953) relative permeability function is described as

Aqueous phase relative permeability can be computed as a function of aqueous saturation from knowledge of the soil-moisture retention function and the pore distribution model of Burdine [1953]. If the van Genuchten and Brooks and Corey soil-moisture retention functions are used, closed-form expressions for fluid phase relative permeability can be derived. Using the van Genuchten soil-moisture retention function, the aqueous phase relative permeability appears as shown:

Using the Brooks and Corey soil-moisture retention function, the aqueous phase relative permeability appears as shown:

Symbols

aqueous relative permeability
effective aqueous liquid saturation
van Genuchten m parameter
Brooks and Corey parameter
capillary head, m
Mualem

The Mualem(1976) relative permeability function is described as

where L = 0.5.

Aqueous phase relative permeability can be computed as a function of aqueous saturation from knowledge of the soil-moisture retention function and the pore distribution model of Mualem [1976]. If the van Genuchten and Brooks and Corey soil-moisture retention functions are used, closed-form expressions for fluid phase relative permeability can be derived. Using the van Genuchten soil-moisture retention function, the aqueous  relative permeability appears as shown:

Using the Brooks and Corey soil-moisture retention function, the aqueous phase relative permeability appears as shown:

Symbols

aqueous relative permeability
effective aqueous liquid saturation
capillary head, m
van Genuchten m parameter
Brooks and Corey parameter
Modified Mualem

The pore scale parameter (aka tortuosity-connectivity coefficient) can be specified to be any value. The default value is 0.5.

Irreducible Mualem

There is a minimum saturation for the porous medium.

Corey Model

Aqueous- and gas-phase relative permeability can be computed from modified expressions for effective aqueous saturation according to the empirical model of Corey [1977]. The model of Corey accounts for trapped air through a modification to the definition of the effective aqueous saturation according to equation (1). The Corey functions for aqueous- and gas-phase relative permeability are computed according to equations (2) and (3), respectively.

Symbols

aqueous relative permeability
gas relative permeability
effective aqueous liquid saturation
apparent aqueous liquid saturation
actual aqueous liquid saturation
actual aqueous liquid residual saturation
actual gas saturation trapped by aqueous phase
Free Corey

The free Corey is a modified version of the empirical model of Corey [1977]. The model accounts for trapped air through a modification to the definition of the effective aqueous saturation according to equation (1). The free Corey function for aqueous-phase relative permeability is computed according to equation (2).

Symbols

apparent aqueous liquid saturation
effective aqueous liquid saturation
actual aqueous liquid saturation
actual aqueous liquid residual saturation
actual gas saturation trapped by aqueous phase
aqueous relative permeability
Free Corey model parameter
Free Corey model parameter
Haverkamp

Symbols

aqueous relative permeability
capillary head, m
Haverkamp function parameter
Haverkamp function parameter
Touma and Vauclin

Symbols

aqueous relative permeability
effective aqueous liquid saturation
Touma and Vauclin function parameter
 Touma and Vauclin function parameter
Tabular

This option accepts tabulated relative permeability data. The default is the data of aqueous saturation and aqueous relative permeability data pairs and linear interpolation is used between data points; a cubic spline interpolation scheme can also be specified. Alternately, capillary head vs relative permeability can be provided using the keyword "head." Other interpolation schemes that can be specified for capillary head vs. aqueous relative permeability are log capillary head vs relative permeability, cubic spline, or cubic spline for log capillary head vs relative permeability.

Sub-Options

Dual Porosity/Permeability Model for Fractured Systems

 Dual porosity functions or equivalent continuum models [Klavetter and Peters 1986; Nitao 1988] relate bulk fluid phase relative permeabilities to those for the fracture and matrix according to equations (1) and (2). Dual porosity models require the assumption that fracture and matrix fluid pressures are in equilibrium, which inherently neglects transient fracture-matrix interactions. Fracture and matrix relative permeabilities are computed from either the Burdine or Mualem models using either the van Genuchten or Brooks and Corey soil moisture retention functions. In these functions the effective aqueous and gas saturations are replaced with the corresponding values for the fracture and matrix components of the soil. For example, the fracture and matrix aqueous relative permeabilities for the Burdine model with the Brooks and Corey soil-moisture retention function are shown in equations (3) and (4), respectively.

  Symbols

 bulk aqueous relative permeability
 bulk gas relative permeability
 intrinsic permeability of the fracture material, m2
 intrinsic permeability of the matrix material, m2
aqueous relative permeability of the fracture material
 aqueous relative permeability of the matrix material
gas relative permeability of the fracture material 
gas relative permeability of the matrix material 
 effective aqueous liquid saturation of the fracture material
effective aqueous liquid saturation of the matrix material 
 diffusive porosity of the fracture material
 diffusive porosity of the matrix material
 Brooks and Corey parameter

IJK, JKI or KIJ Indexing

If IJK Indexing, JKI Indexing, or KIJ Indexing is specified as the Rock/Soil Name in the Rock/Soil Zonation Card, then this must also be specified as the Rock/Soil Name in the Directional Aqueous Relative Permeability Card.

IJK Indexing

If the IJK Indexing option is specified in the Rock/Soil Zonation Card, then the relative permeability models and any or all associated parameters can be specified either as a single value that will be applied to each node in the domain, or in an external file with the values for every grid-cell ordered according to the IJK indexing scheme. Units shown in the input line will be applied to all parameters in the external file. 

JKI Indexing

If the JKI Indexing option is specified in the Rock/Soil Zonation Card, then the relative permeability models and any or all associated parameters can be specified either as a single value that will be applied to each node in the domain, or in an external file with the values for every grid-cell ordered according to the JKI indexing scheme. Units shown in the input line will be applied to all parameters in the external file. 

KIJ Indexing

If the KIJ Indexing option is specified in the Rock/Soil Zonation Card, then the relative permeability models and any or all associated parameters can be specified either as a single value that will be applied to each node in the domain, or in an external file with the values for every grid-cell ordered according to the KIJ indexing scheme. Units shown in the input line will be applied to all parameters in the external file. 

Effective Aqueous Liquid Saturation

Mobile saturations are scaled by the residual aqueous liquid saturation to determine effective saturations for use with common relative permeability models. 

Symbols

effective aqueous liquid saturation
actual aqueous liquid saturation
 actual aqueous liquid residual saturation


References

Burdine, NT. 1953. "Relative Permeability Calculation from Size Distribution Data," Trans. AIME, 198:71-78.

Corey, AT. 1977. Mechanics of Heterogeneous Fluids in Porous Media, Fort Collins, Colorado.

Fatt , I and WA Klikoff Jr. 1959. "Effect of Fractional Wettability on Multiphase Flow through Porous Media," AIME Trans, 215:426-429.

Gardner, WR. 1958. "Some Steady-State Solutions of the Unsaturated Moisture Flow Equation with Applications to Evaporation from a Water Table," Soil Science, 85:228-232.

Haverkamp, R, M Vauclin, J Touma, PJ Wierenga, and G Vachaud. 1977. "A Comparison of Numerical Simulation Models for One-Dimensional Infiltration," Soil Sci. Soc. Am. J., 41:285-294.

Hoffmann-Riem, H, MT Van Genuchten, and H Fluhler. 1999. A General Model of the Hydraulic Conductivity of Unsaturated Soils. In M.T. Van Genuchten, L.J. Leij and L. Wu Proceedings of Int. Workshop, Characterization And Measurements Of The Hydraulic Properties of UnsaturatedPorous Media, University of California Riverside, Riverside, CA.

Klavetter, EA and RR Peters. 1986. Estimation of Hydrologic Properties of Unsaturated Fractured Rock Mass, SAND84-2642, Sandia National Laboratories, Albuquerque, NM.

Mualem, Y. 1976. "A New Model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media," Water Resources Research, 12:513-522.

Nitao, JJ. 1988. Numerical Modeling of the Thermal and Hydrological Environment around a Nuclear Waste Package Using the Equivalent Continuum Approximation: Horizontal Emplacement, UCID-2144, Lawrence Livermore National Laboratory, Livermore, CA.

Parker, JC and RJ Lenhard. 1987. "A Model for Hysteretic Constitutive Relations Governing Multiphase Flow 1. Saturation-Pressure Relations," Water Resources Research, 23(12):2187-2196.

Polmann, DJ. 1990. "Application of Stochastic Methods to Transient Flow and Transport in Heterogeneous Unsaturated Soils," PhD, Massachusetts Institute of Technology, Cambridge, MA.

Raats, PaC, ZF Zhang, AL Ward, and GW Gee. 2004. "The Relative Connectivity–Tortuosity Tensor for Conduction of Water in Anisotropic Unsaturated Soils," Vadose Zone Journal, 3:1471-1478.

Zhang, ZF, AL Ward, and GW Gee. 2003. "A Tensorial Connectivity–Tortuosity Concept to Describe the Unsaturated Hydraulic Properties of Anisotropic Soils," Vadose Zone Journal, 2:313-321.

 

 

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