Skip to main content

 

  • About
  • News & Media
  • Careers
  • Events

Breadcrumb

STOMP User Guide

  • STOMP Website
  • Introduction
    • Operational Mode Specific Input Guides
    • Availability and Licensing
    • Installation
  • Fundamentals of STOMP
    • Code Design
    • Numerical Solutions
    • Description of Variables
    • List of Variable Names
    • Accepted Units
    • Glossary of Symbols
  • Using STOMP
    • Pre-Processing
    • Input File Structure
    • Generated Output Files
    • Post-Processing
  • Example Short Course Problems
  • Additional User Documentation

Breadcrumb

  1. STOMP User Guide
  2. STOMP Input

Saturation Function Card Options

Saturation functions relate the gas-aqueous capillary pressure to aqueous, gas, and entrapped gas saturations. Model options and parameters for these functions are specified through the Saturation Function Card. Every rock/soil type defined on the Rock/Soil Zonation Card must be referenced. With the IJK Indexing option, node dependent parameters are entered via external files and node independent parameters are entered directly on the card. Functional forms for the saturation-capillary pressure functions are preferred; however, tabular input is acceptable. By default, tabular data will be interpolated using linear interpolation, whereas values beyond the table limits will be assigned either the table minimum or maximum values appropriately.

  • van Genuchten
  • Brooks and Corey
  • Haverkamp
  • Gas Entrapment
  • Webb Extension
  • Fractured Media
  • Tabular Input

van Genuchten Functional Form

This option uses the van Genuchten (1980) retention function.

van Genuchten equation

where

van Genuchten m
Symbols
effective aqueous liquid saturation symbol effective aqueous liquid saturation
gas pressure symbol gas pressure, Pa
aqueous pressure symbol aqueous liquid pressure, Pa
1/m van Genuchten function parameter, 1/m
n van Genuchten n parameter
m van Genuchten m parameter
reference aqueous density reference aqueous density, kg/m3
g acceleration of gravity, m/s2

Brooks and Corey Functional Form

The Brooks and Corey (1966) retention function is used.

Brooks and Corey equation
Symbols
effective aqueous liquid saturation symbol effective aqueous liquid saturation
gas pressure symbol gas pressure, Pa
aqueous pressure symbol aqueous liquid pressure, Pa
reference aqueous density reference aqueous density, kg/m3
g acceleration of gravity, m/s2
lambda Brooks and Corey lambda parameter
psi Brooks and Corey function nonwetting fluid entry head, m

Haverkamp Functional Form

The Haverkamp et al. (1977) retention function is used.

Haverkamp equation

Symbols
theta volumetric water content
theta r residual water content
theta s saturated water content
capillary head capillary head, m
alpha Haverkamp α parameter
beta Haverkamp β parameter

Gas Entrapment

This option considers entrapped gas that is immobile and also reduces the pore fraction for the mobile fluid.

A theoretical model for hysteretic saturation functions for aqueous-gas systems was developed by Parker and Lenhard [1987]. A simplified version of this model, analogous to Kaluarachchi and Parker [1992], has been implemented in the STOMP simulator. The model includes effects of gas entrapment during aqueous-phase imbibition paths. Gas entrapment during aqueous-phase imbibition will depend on the aqueous saturation and the current saturation path. The amount of entrapped gas varies linearly between zero and the gas effective residual saturation with the apparent saturation, which varies between the reversal point from main drainage to one. Gas effective residual saturations are computed using an empirical relationship developed by Land [1968] for aqueous-NAPL systems. In this simplified hysteretic model for aqueous-gas systems, gas can be trapped or free, where free gas refers to continuous volumes which advect freely, and trapped gas refers to discontinuous ganglia of gas occluded within the aqueous phase. Occluded gas is assumed to be immobile. The apparent aqueous saturation equals the effective aqueous saturation plus effective trapped gas saturation, as shown in the Equation 1 below. The effective gas saturation equals the effective trapped and free gas saturations, as shown in Equation 3. In hysteretic systems, the residual saturation is independent of capillary pressure.

entrapment equation

where

entrapment equation

The saturation functions relate gas-aqueous capillary pressure to apparent aqueous saturations, according to equations (4) and (5), for the van Genuchten and Brooks and Corey functions, respectively. The effective trapped gas saturation is computed according to equation (7), which recognizes that entrapped gas cannot exceed the gas present. Land’s parameter for gas-aqueous interfaces is computed according to equation (8).

entrapment equation

where

entrapment equation
Symbols
Sl apparent aqueous liquid saturation
Sl effective aqueous liquid saturation
Sgl effective gas saturation trapped by aqueous phase
Sl actual aqueous liquid saturation
Slr actual aqueous liquid residual saturation
Sg effective gas saturation
Sgf effective free gas saturation
Sg actual gas saturation
alpha van Genuchten function parameter, 1/m
n van Genuchten n parameter
m van Genuchten m parameter
Pg gas pressure, Pa
Pl aqueous liquid pressure, Pa
Pl reference aqueous density, kg/m3
g acceleration of gravity, m/s2
psi Brooks and Corey function nonwetting fluid entry head, m
lambda Brooks and Corey parameter
slmin minimum effective aqueous liquid saturation
sgrmax maximum effective residual gas saturation
Lg  Land's parameter for gas-aqueous interface

Webb Extension

To extend the saturation function below the aqueous residual saturation, functional extensions are required. STOMP-CO2 and -CO2E recognize the Webb (2000) model. This function works with both the van Genuchten and Brooks and Corey characteristics functions, without requiring additional input as the oven-dried head is assumed to be equal to a capillary pressure of 109 Pa (~105 m).

Fractured Media (Equivalent Continuum)

This option is used for a fractured medium. The retention properties for both the fracture and the matrix are needed. Dual porosity functions or equivalent continuum models for aqueous-gas systems relate the gas-aqueous capillary pressure to the bulk aqueous saturation for fractured geologic media through two functions [Klavetter and Peters 1986; Nitao 1988]. One function relates the gas aqueous capillary pressure to the matrix aqueous saturation and the other relates the gas-aqueous capillary pressure to the fracture aqueous saturation. The pivotal assumption associated with the dual porosity function is that the fracture and matrix pressures are in equilibrium. This assumption neglects transient fracture-matrix interactions. Fracture and matrix effective saturations can be computed with either van Genuchten or Brooks and Corey functions above. The bulk aqueous saturation is computed by combining the fracture and matrix aqueous saturations and diffusive porosities:

equivalent continuum equation

where the actual saturations are computed from effective saturations, according to

actual saturations

Symbols
actual aqueous liquid saturation symbol actual aqueous liquid saturation
actual aqueous liquid saturation symbol actual aqueous liquid saturation of the fracture material
actual aqueous liquid saturation symbol actual aqueous liquid saturation of the matrix material
actual aqueous liquid residual saturation  symbol actual aqueous liquid residual saturation of the fracture material
actual aqueous liquid residual saturation symbol actual aqueous liquid residual saturation of the matrix material
effective aqueous liquid saturation symbol effective aqueous liquid saturation of the fracture material
effective aqueous liquid saturation symbol effective aqueous liquid saturation of the matrix material
diffusive porosity symbol diffusive porosity of the fracture material
diffusive porosity symbol diffusive porosity of the matrix material

Tabular Input

This option accepts tabulated retention data. The default is the data of pressure head and saturation data pairs and linear interpolation is used between data points. Alternately, water content vs capillary head can be provided using the keyword "water content." Other interpolation schemes that can be specified are linear-log, cubic spline, or cubic-spline-log.

tip icon References

Brooks, RH and AT Corey. 1966. "Hydraulic Properties of Porous Media Affecting Fluid Flow," Proc. ASCE J. Irrig. Drain. Div. , 92:61-68.

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.

Kaluarachchi, JJ and JC Parker. 1992. "Multiphase Flow with a Simplified Model for Oil Entrapment," Transport in Porous Media, 7:1-14.

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

Land, CS. 1968. "Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow from Rock Properties," Trans. Am. Inst. Min. Metall. Pet. Eng., 243:149-156.

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.

van Genuchten, MT. 1980. "A Closed-Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils," Soil Science Society of America Journal, 44:892-898.

Webb, S. W., 2000. “A simple extension of two-phase characteristic curves to include the dry region.” Water Resources Research, 36(6):1425-1430.

PNNL

  • Get in Touch
    • Contact
    • Careers
    • Doing Business
    • Security & Privacy
  • Research
    • Scientific Discovery
    • Energy Resiliency
    • National Security
Sign up for our newsletter
Department of Energy Logo Battelle Logo
Pacific Northwest National Laboratory (PNNL) is managed and operated by Battelle for the Department of Energy
  • YouTube
  • Facebook
  • Twitter
  • Instagram
  • LinkedIn