HYDRATE Mode

Governing Equations

The HYDRATE mode involves two components (water and gas), four phases (liquid, gas, hydrate, and ice) coupled to energy conservation. Mass conservation equations have the form

(1)\[\frac{{{\partial}}}{{{\partial}}t} \varphi \sum_{{{\alpha}}=l,\,g,\,h,\,i} \Big(s_{\alpha}^{} \rho_{\alpha}^{} x_j^{\alpha} \Big) + {\boldsymbol{\nabla}}\cdot\Big({\boldsymbol{q}}_l^{} \rho_l^{} x_j^l + {\boldsymbol{q}}_g \rho_g^{} x_j^g -\varphi s_l^{} D_l^{} \rho_l^{} {\boldsymbol{\nabla}}x_j^l -\varphi s_g^{} D_g^{} \rho_g^{} {\boldsymbol{\nabla}}x_j^g \Big) = Q_j^{},\]

for liquid, gas, hydrate, and ice saturations \(s_{l,\,g,\,h,\,i}^{}\), mobile phase density \(\rho_{l,\,g}^{}\), diffusivity \(D_{l,\,g}^{}\), Darcy velocity \({\boldsymbol{q}}_{l,\,g}^{}\) and liquid and gas mole fractions \(x_j^{l,\,g}\). Mole fractions of components in the hydrate and ice phases are fixed. The energy conservation equation can be written in the form

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

as the sum of contributions from liquid and fluid phases and solid hydrate, ice, and rock phases; 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 \(\kappa\) can be determined from THERMAL_CHARACTERISTIC_CURVES, or through the HYDRATE block in HYDRATE :

(4)\[\kappa = \kappa_{\rm dry} + {\phi}\sum_{{{\alpha}}=l,\,g,\,h,\,i}s_{{\alpha}}{\kappa}_{\alpha} ,\]

where \(\kappa_{\rm dry}\) is the dry thermal conductivity, an input parameter equivalent to \((1-\phi)\kappa_{\rm rock}\) and \(\kappa_{\rm sat}\) are dry and fully saturated rock thermal conductivities, respectively.

The internal energy of the ice phase is computed by (Fukusako and Yamada, 1993):

(5)\[U_i = L_w + 185T + 3.445(T^2 - 273.15^2 ) , T >= 90K\]

(6)\[U_i = L_w + 4.475(T^2 - 273.15^2 ), T < 90K,\]

where T is the temperature and \(L_w\) is the latent heat ofo fusion for water, set at -6017.1 J/mol.

The enthalpy of the hydrate phase is computed by (Handa, 1998):

(7)\[H_h = C_{ph}(T-T_f)+H_{h0}/(N_{hyd}+1)\]

where \(H_{h0}\) is the hydrate reference enthalpy, set to -54.734 kJ/mol, \(N_{hyd}\) is the hydration number (set to 6), \(T_{f}\) is the freezing point (0C in the absence of salt), and \(C_{ph}\) is the heat capacity of hydrate, computed by:

(8)\[C_{ph} = 1620(M_w N_{hyd} + M_m)/1000\]

where \(M_w\) is the molecular weight of water and \(M_m\) is the molecular weight of methane

Two-phase liquid-gas equilibrium partitioning is computed via Henry’s Law (Carroll and Mather, 1997):

(9)\[K_h = 1000e^{5.1345+7837/T - 1.509x10^6/T^2 + 2.06x10^7/T^3}\]

The liquid-hydrate phase boundary is computed by (Moridis, 2003):

(10)\[P_e = e^{0.0334940999T-8.1938174346}, T < T_f\]

(11)\[P_e = e^{0.1100383278T-29.1133440975}, T >= T_f\]

where \(P_e\) is the 3-phase equilibrium pressure in MPa.

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

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

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

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

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

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

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 RICHARDS Mode. 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 RICHARDS Mode becomes the effective liquid saturation \(s_{el}\) in the multiphase formulation. Liquid saturation \(s_l\) is obtained from the effective liquid saturation by

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

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

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

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

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

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

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

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

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

Relative Permeability Functions

Two forms of each relative permeability function are implemented based on the Mualem and Burdine formulations as in RICHARDS Mode, 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 \(1/2\). If the keyword EFFECTIVE_SATURATION_SCALING is invoked in the HYDRATE block, then each mobile phase (e.g., liquid or gas) saturation is scaled by the sum of mobile phase saturations. For example, if liquid and gas saturations are each 30%, their effective phase satuations each become 50%. 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

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

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

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

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

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

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

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

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

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

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

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