# Mode: `GENERAL`

¶

The `GENERAL`

mode involves two phase liquid water-gas flow coupled
to the reactive transport mode. Mass conservation equations have the
form

for liquid and gas saturation \(s_{l,\,g}^{}\), density \(\rho_{l,\,g}^{}\), diffusivity \(D_{l,\,g}^{}\), Darcy velocity \({\boldsymbol{q}}_{l,\,g}^{}\) and mole fraction \(x_i^{l,\,g}\). The energy conservation equation can be written in the form

as the sum of contributions from liquid and gas fluid phases and rock, with internal energy \(U_{{\alpha}}\) and enthalpy \(H_{{\alpha}}\) of fluid phase \({{\alpha}}\), rock heat capacity \(C_p\) and thermal conductivity \(\kappa\). Note that

Thermal conductivity \(\kappa\) is determined from the equation (Somerton et al., 1974)

where \(\kappa_{\rm dry}\) and \(\kappa_{\rm sat}\) are dry and fully saturated rock thermal conductivities.

The Darcy velocity of the \(\alpha^{th}\) phase is equal to

where \(\boldsymbol{g}\) denotes the acceleration of gravity, \(k\) denotes the saturated permeability, \(k^{r}_{\alpha}\) the relative permeability, \(\mu_\alpha\) the viscosity, \(p_\alpha\) the pressure of the \(\alpha^{th}\) fluid phase, and

with \(W_\alpha\) the gram formula weight of the \(\alpha^{th}\) phase

where \(W_i\) refers to the formula weight of the \(i^{th}\) component.

## Capillary Pressure - Saturation Functions¶

Capillary pressure is related to effective liquid saturation by the van Genuchten and Brooks-Corey relations, as described under the sections van Genuchten Saturation Function and Brooks-Corey Saturation Function under Mode: RICHARDS. Because both a liquid (wetting) and gas (non-wetting) phase are considered, the effective saturation \(s_e\) in the van Genuchten and Brooks-Corey relations under Mode: RICHARDS becomes the effective liquid saturation \(s_{el}\) in the multiphase formulation. Liquid saturation \(s_l\) is obtained from the effective liquid saturation by

where \(s_{rl}\) denotes the liquid residual saturation, and \(s_0\) denotes the maximum liquid saturation. The gas saturation can be obtained from the relation

The effective gas saturation \(s_{eg}\) is defined by the relation

Additionally, a linear relationship between capillary pressure \(p_c\) and effective liquid saturation can be described as

where \(\alpha\) is a fitting parameter representing the air entry pressure [Pa]. The inverse relationship for capillary pressure is

## Relative Permeability Functions¶

Two forms of each relative permeability function are implemented based on the Mualem and Burdine formulations as in Mode: RICHARDS, but the effective liquid saturation \(s_{el}\) and the effective gas saturation \(s_{eg}\) are used. A summary of the relationships used can be found in Chen et al. (1999), where the tortuosity \(\eta\) is set to \(1/2\). The implemented relative permeability functions include: Mualem-van Genuchten, Mualem-Brooks-Corey, Mualem-linear, Burdine-van Genuchten, Burdine-Brooks-Corey, and Burdine-linear. For each relationship, the following definitions apply:

For the Mualem relative permeability function based on the van Genuchten saturation function, the liquid and gas relative permeability functions are given by the expressions

For the Mualem relative permeability function based on the Brooks-Corey saturation function, the liquid and gas relative permeability functions are given by the expressions

For the Mualem relative permeability function based on the linear saturation functions, the liquid and gas relative permeability functions are given by the expressions

For the Burdine relative permeability function based on the van Genuchten saturation function, the liquid and gas relative permeability functions are given by the expressions

For the Burdine relative permeability function based on the Brooks-Corey saturation function, the liquid and gas relative permeability functions have the form

For the Burdine relative permeability function based on the linear saturation functions, the liquid and gas relative permeability functions are given by the expressions

## Kelvin’s Equation for Vapor Pressure Lowering¶

Vapor pressure lowering resulting from capillary suction is described by Kelvin’s equation given by

where \(p_v\) represents the vapor pressure, \(p_{\rm sat}\) the saturation pressure of pure water, \(p_c\) capillary pressure, \(\rho_l\) liquid mole density, \(T\) denotes the temperature, and \(R\) the gas constant.