Aqueous Relative Permeability Options (e-W)
Relative Permeability Options
Constant
The relative permeability is constant regardless of saturation.
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Mualem
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Modified Mualem
The pore scale parameter (aka tortuosity-connectivity coefficient) can be specified to be any value. The default value is 0.5.
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Irreducible Mualem
There is a minimum saturation for the porous medium.
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Mualem with Polmann
The Mualem (1976) relative permeability model is used to calculate the vertical permeability and the Polmann (1990) model is used to calculate horizontal permeability.
Polmann (1990) used a stochastic model to evaluate tension-dependent anisotropy. The Gardner (1958) relationship was used to describe unsaturated hydraulic conductivity as a function of saturated hydraulic conductivity and tension. Using a linear correlation model between the zero-tension intercept and β, Polmann (1990) presented a generalized model that accounts for the cross-correlation of the local soil property (i.e., lnKs and β) residual fluctuations. Compared against the uncorrelated lnKs and β model, the partial correlation of the properties was shown to have a significant impact on the magnitude of the effective parameters derived from the stochastic theory. The Polmann (1990) equations for deriving the effective parameters for strongly stratified porous media are as shown below.
A modified version of the original Polmann model is implemented in eSTOMP. This version uses a single segment of the log-linear K(h) function. Because the Polmann model is based on the log-linear hydraulic function, it may be applicable only to a relatively narrow pressure head range over which the parameters are derived. Extrapolating the derived hydraulic function to other pressure head conditions may cause significant error or even physically incorrect results (e.g., the predicted hydraulic conductivity in the horizontal direction may increase with decreasing saturation, which is physically incorrect). It calculates an anisotropy coefficient (Can) and the hydraulic conductivity in the horizontal direction (Kh). The modified model also requires user specified maximum (Canmax) and minimum (Canmin) limits of the anisotropy coefficient. If computed values of Can are outside these limits, the values are truncated as shown in the equations below:
Symbols
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equivalent unsaturated hydraulic conductivity in the horizontal direction, m/s |
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equivalent unsaturated hydraulic conductivity in the vertical direction, m/s |
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unsaturated hydraulic conductivity (pressure dependent), m/s |
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saturated hydraulic conductivity, m/s |
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mean of LnKs |
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capillary head, m |
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standard deviation of the residuals in the LnKs vs. β regression for all the samples considered |
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standard deviation of LnKs |
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variance of LnKs |
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slope between LnKs and h |
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slope of LnKs vs. β regression line for the samples considered |
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mean slope of β for LnK vs. h regression |
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vertical correlation length for LnKs |
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anisotropy coefficient |
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maximum value of anisotropy coefficient |
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minimum value of anisotropy coefficient |
The Directional Aqueous Relative Permeability Card can be used to represent anisotropy for several aqueous relative permeability models. Zhang et al. 2003 and Raats et al. 2004 demonstrate the use of Directional Aqueous Relative Permeability with the Modified Mualem model.
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Fatt and Klikoff
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Corey Model
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Free Corey
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Haverkamp
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Touma and Vauclin
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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.
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Sub-Options
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
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effective aqueous liquid saturation |
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actual aqueous liquid saturation |
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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.