Mode: GENERAL

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

(1)\[\frac{{{\partial}}}{{{\partial}}t} \varphi \Big(s_l^{} \rho_l^{} x_i^l + s_g^{} \rho_g^{} x_i^g \Big) + {\boldsymbol{\nabla}}\cdot\Big({\boldsymbol{q}}_l^{} \rho_l^{} x_i^l + {\boldsymbol{q}}_g \rho_g^{} x_i^g -\varphi s_l^{} D_l^{} \rho_l^{} {\boldsymbol{\nabla}}x_i^l -\varphi s_g^{} D_g^{} \rho_g^{} {\boldsymbol{\nabla}}x_i^g \Big) = Q_i^{},\]

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

(2)\[\sum_{{{\alpha}}=l,\,g}\left\{\frac{{{\partial}}}{{{\partial}}t} \big(\varphi s_{{\alpha}}\rho_{{\alpha}}U_{{\alpha}}\big) + {\boldsymbol{\nabla}}\cdot\big({\boldsymbol{q}}_{{\alpha}}\rho_{{\alpha}}H_{{\alpha}}\big) \right\} + \frac{{{\partial}}}{{{\partial}}t} \Big((1-\varphi)\rho_r C_p T \big) - {\boldsymbol{\nabla}}\cdot (\kappa{\boldsymbol{\nabla}}T)\Big) = Q,\]

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

(3)\[U_{{\alpha}}= H_{{\alpha}}-\frac{P_{{\alpha}}}{\rho_{{\alpha}}}.\]

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

(4)\[\kappa = \kappa_{\rm dry} + \sqrt{s_l^{}} (\kappa_{\rm sat} - \kappa_{\rm dry}),\]

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

(5)\[\boldsymbol{q}_\alpha = -\frac{k k^{r}_{\alpha}}{\mu_\alpha} (\boldsymbol{\nabla} p_\alpha - \gamma_\alpha \boldsymbol{g}), \ \ \ (\alpha=l,g),\]

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

(6)\[\gamma_\alpha^{} = W_\alpha^{} \rho_\alpha^{},\]

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

(7)\[W_\alpha = \sum_{i=w,\,a} W_i^{} x_i^\alpha,\]

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

(8)\[s_{l} = s_{el}s_0 - s_{el}s_{rl} + s_{rl},\]

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

(9)\[s_l + s_g = 1\]

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

(10)\[s_{eg} = 1 - \frac{s_l-s_{rl}}{1-s_{rl}-s_{rg}}\]

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

(11)\[s_{el} = {{p_c-p_c^{max}}\over{\frac{1}{\alpha}-p_c^{max}}}\]

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

(12)\[p_c = \left({\frac{1}{\alpha}-p_c^{max}}\right)s_{el} + p_c^{max}\]

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:

\[ \begin{align}\begin{aligned}S_{el} = \frac{S_{l}-S_{rl}}{1-S_{rl}}\\S_{eg} = \frac{S_{l}-S_{rl}}{1-S_{rl}-S_{rg}}\end{aligned}\end{align} \]

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

(13)\[ \begin{align}\begin{aligned}k^{r}_{l} =& \sqrt{s_{el}} \left\{1 - \left[1- \left( s_{el} \right)^{1/m} \right]^m \right\}^2\\k^{r}_{g} =& \sqrt{1-s_{eg}} \left\{1 - \left( s_{eg} \right)^{1/m} \right\}^{2m}.\end{aligned}\end{align} \]

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

(14)\[ \begin{align}\begin{aligned}k^{r}_{l} =& \big(s_{el}\big)^{5/2+2/\lambda}\\k^{r}_{g} =& \sqrt{1-s_{eg}}\left({1-s_{eg}^{1+1/\lambda}}\right)^{2}.\end{aligned}\end{align} \]

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

(15)\[ \begin{align}\begin{aligned}k^{r}_{l} =& \sqrt{s_{el}}\frac{\ln\left({p_c/p_c^{max}}\right)}{\ln\left({\frac{1}{\alpha}/p_c^{max}}\right)}\\k^{r}_{g} =& \sqrt{1-s_{eg}}\left({1-\frac{k^{r}_{l}}{\sqrt{s_{eg}}}}\right)\end{aligned}\end{align} \]

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

(16)\[ \begin{align}\begin{aligned}k^{r}_{l} =& s_{el}^2 \left\{1 - \left[1- \left( s_{el} \right)^{1/m} \right]^m \right\}\\k^{r}_{g} =& (1-s_{eg})^2 \left\{1 - \left( s_{eg} \right)^{1/m} \right\}^{m}.\end{aligned}\end{align} \]

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

(17)\[ \begin{align}\begin{aligned}k^{r}_{l} =& \big(s_{el}\big)^{3+2/\lambda}\\k^{r}_{g} =& (1-s_{eg})^2\left[{1-(s_{eg})^{1+2/\lambda}}\right].\end{aligned}\end{align} \]

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

(18)\[ \begin{align}\begin{aligned}k^{r}_{l} =& s_{el}\\k^{r}_{g} =& 1 - s_{eg}.\end{aligned}\end{align} \]

Kelvin’s Equation for Vapor Pressure Lowering

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

(19)\[p_v = p_{\rm sat} (T) e^{-p_c/\rho_l RT},\]

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.