STOMP

Aqueous Relative Permeability Card Options (CO2E)

Effective Saturations

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

 

The effective aqueous liquid saturation:
The effective aqueous nonaqueous liquid saturation:
The effective gas saturation:
The wetting phases are combined for a total effective liquid saturation:

Relative Permeability Models

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:

 

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:

 

Dual Porosity Relative Permeability Functions:

Dual porosity functions or equivalent continuum models [Klavetter and Peters 1986; Nitao 1988] relate bulk fluid phase relative permeabilities to the those for the fracture and matrix according to the equation below.

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 below.

 

Fatt and Klikoff:

the Fatt and Klikoff (1959) model

Corey:

Aqueous- and gas-phase relative permeabilities can be computed from modified expressions for effective aqueous saturation according to the empirical model of Corey [1977]. The Corey's curves account for trapped air through a modification to the definition of the effective aqueous saturation according to:

The Corey function for aqueous phase relative permeability is computed according to:

Free Corey:

The free Corey model is a modified version of the empirical Corey [1977] model. The model accounts for trapped air through a modification to the definition of the effective aqueous saturation according:

The free Corey function for aqueous-phase relative permeability is computed according to:

Haverkamp:

the Haverkamp et al. (1977) model

Tauma and Vauclin:

the Tauma and Vauclin model

Mualem with Polmann:

The Mualem (1976) relative permeability model is used to calculate the vertical permeability; 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 K0 and ag, Polmann (1990) presented a generalized model that accounts for the cross-correlation of the local soil property (i.e., lnK0 and ag) residual fluctuations.  Compared against the uncorrelated lnK0 and ag 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) equation for deriving the effective parameters for strongly stratified porous media are is shown below.

 

Warning

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).

The Polmann model was modified when it was incorporated into the STOMP (White et al. 2001; White and Oostrom 2006) numerical flow simulator.  It was used to calculate the anisotropy coefficient (C), and hence, the hydraulic conductivity in the horizontal direction (Khmsa) is determined based on the user-defined hydraulic conductivity in the vertical direction (Kv).  Furthermore, the maximum (Cmax) and minimum (Cmin) limits of the anisotropy coefficient are enforced.  The definitions of C and Khmsa associated with the Polmann model in STOMP are given below.

 

Irreducible Mualem:
there is a minimum saturation for the porous medium.
Tabular:

Tabulated relative permeability.

Choose One

  The user may input data in any of the following formats.

  1. Aqueous relative permeability versus aqueous saturation
  2. Aqueous relative permeability versus ln(capillary head)
  3. Aqueous relative permeability versus capillary head Interpolation

Choose One

  The user must select an interpolation scheme for the data.

  1. Linear interpolation and table truncation beyond limits
  2. Cubic spline interpolation

Rijtema-Gardner modified exponential function:

The Rijtema-Gardner modified exponential function [1965] is based on Darcy's law and the modified Gardner's [1958] exponential conductivity equation:

 

 

References

Klavetter, E.A., Peters, R.R., 1986. Estimation of hydrologic properties of unsaturated fractured rock mass. Sandia National Laboratories, Albuquerque, NM.

Nitao, J.J., 1988. Numerical modeling of the thermal and hydrological environment around a nuclear waste package using the equivalent continuum approximation: horizontal emplacement. Lawrence Livermore National Laboratory, Livermore, CA.

Burdine, N.T., 1953. Relative permeability calculation from size distribution data. Trans. AIME 198, 71-78. 

Haverkamp, R., M. Vauclin, J. Touma, P.J. Wierenga and G. Vachaud (1977). A comparison of numerical simulation models for one-dimensional infiltration. Soil Science Society of America Journal 41, 285-294.

Mualem, Y., 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research 12, 513-522.

Corey, A.T., 1977. Mechanics of heterogeneous fluids in porous media. Water Resources Publications, Fort Collins, Colorado.

Fatt , I., Klikoff Jr., W.A., 1959. Effect of fractional wettability on multiphase flow through porous media. AIME Trans 215, 426-429. 

Rijtema, P.E., 1965. An analysis of actual evapotranspiration. Center for Agricultural Publications and Documentation.

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

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

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