A clear understanding of the terms component and phase is necessary when using the TOUGH codes. Consider a system consisting of water and air (implemented as EOS3 in TOUGH2). This system consists of two components (water and air) and will have two phases (liquid and gas). Note that TOUGH2 does not include a solid phase, which would consist of ice; TOUGH-Fx/HYDRATE does include ice as a solid phase.

Importantly, the two components (water and air) can be present in both phases. The liquid phase can consist of liquid water and dissolved air. Similarly, the gaseous phase can be comprised of gaseous air and water vapor.

For single phase conditions, the thermodynamic state is defined by pressure, temperature, and air mass fraction. If the single phase is liquid, then the air mass fraction will be the air dissolved in the water, which is a small value. An example of a valid initial condition specification for single phase liquid is shown in Figure 3.1, with pressure of 1.0E5 Pa, temperature of 20 C, and a small air mass fraction of 1.0E-5. This small amount of air will be dissolved in the water.

If the single phase is gas, the gas can consist of both water vapor and air. The air mass fraction can be as large as 1. A valid initial condition specification for single phase gas is given in Figure 3.2, with pressure of 1.0E5 Pa, temperature of 20 C, and a air mass fraction of 0.999. This means that a small amount of the gas will consist of water vapor.

For two phase conditions, the thermodynamic state is defined by gas phase pressure, gas saturation, and temperature. An example of a two phase initial condition is given in Figure 3.3, with pressure of 1.0E5 Pa, temperature of 20 C, and gas saturation of 0.5.

Figure 3.1. Single phase liquid initial conditions for EOS3

Figure 3.2. Single phase gas initial conditions for EOS3

Figure 3.3. Two phase initial conditions for EOS3

As an example, a single element with a volume of 1 cubic meter and 0.1 porosity was run using the initial conditions given above. The resulting solution and mass fractions for each component are given below, in Figure 3.4 and Figure 3.5. The two phase solution is of particular interest since this case provides the saturation conditions for water vapor in the gas phase and dissolved air in the liquid phase. The reader is encouraged to use single element problems when starting to use a new equation of state (EOS).

Figure 3.4. States corresponding to the three initial condition options

Figure 3.5. States corresponding to the three initial condition options

In TOUGH2, all water properties are represented by the steam table equations as given by the International Formulation Committee [International Formulation Committee, 1987]. Air is approximated as an ideal gas, and addititivity is assumed for air and vapor partial pressures in the gas phase. The viscosity of air-vapor mixtures is computed from a formulation give by Hirshfelder et al. [Hirschfelder et al., 1954]. The solubility of air in liquid water is represented by Henry's law.

Because of the detailed physics that are included in the TOUGH codes, setting of multi-phase initial conditions requires detailed understanding of the problem. For help in setting mixture conditions, the user may refer to a thermodynamics text, such as [Cengel and Boles, 1989].

As described in the TOUGH2 manual, the basic mass and energy balance equations solved by TOUGH2 can be written in the general form:

The integration is over an arbitrary subdomain of the flow system under study, which is bounded by the closed surface . The quantity appearing in the accumulation term (left hand side) represents mass or energy per volume, with labeling the mass components and an extra heat "component" if the analysis is nonisothermal. denotes mass or heat flux and denotes sinks and sources. is a normal vector on surface element , pointing inward to .

The user should consult Appendix A of the TOUGH2 User's Guide [Pruess, Oldenburg, and Moridis, 1999] for a further discussion of this topic.

As described in the TOUGH2 User's Manual, the continuum equations are discretized in space using the integral finite difference method (IFD), [Edwards, 1972] and [Narisham and Witherspoon, 1976]. Introducing appropriate volume averages, we have

where is a volume-normalized extensive quantity, and is the average value of over . Surface integrals are approximated as a discrete sum of averages over surface segments :

Here is the average value of the (inward) normal component of over the surface segment between volume elements and . The discretization approach used in the integral finite difference method and the definition of the geometric parameters are illustrated in Figure 3.6.

Figure 3.6. Space discretization and geometry data in the integral finite difference method (from TOUGH2 User's Guide)

The discretized flux is expressed in terms of averages over parameters for elements and . For the basic Darcy flux term, we have

where the subscripts (nm) denote a suitable averaging at the interface between grid blocks n and m (interpolation, harmonic weighting, upstream weighting). is the distance between the nodal points n and m, and is the component of gravitational acceleration in the direction from m to n.

The user should consult Appendix B of the TOUGH2 User's Guide [Pruess, Oldenburg, and Moridis, 1999] for a further discussion of this topic.

Substituting the volume averaged quantities and surface integral approximations into the mass and energy balance, a set of first-order ordinary differential equations in time is obtained.

Time is discretized as a first order finite difference, and the flux and sink and source terms on the right-hand side are evaluated at the new time, to obtain the numerical stability needed for an efficient calculation of multiphase flow.

The user should consult Appendix B of the TOUGH2 User's Guide [Pruess, Oldenburg, and Moridis, 1999] for a further discussion of this topic.