Concept of flow simulation

Fluid flow of a dissociating or growing hydrate structure can be modeled as a three-phase, two-component flow system. The three phases are gas, water and hydrate, and the two components are gas and water. Only gas and water are flowing phases; hydrate is a solid phase. Pressure reduction of a stable hydrate structure will result in gas production from the dissociating hydrate, thus stimulating fluid flow as a function of time t. The flow of gas and water through a hydrate stability zone, on the other hand, will result in the simulation of hydrate growth. In this first attempt at hydrate simulation, I assume that the system can be modeled under isothermal conditions; thus, only pressure effects are accounted for and any change of temperature will be neglected. In reality, to model such complicated systems as hydrate structures is not a valid assumption. However, it is the most simple model of the problem upon which the more complicated thermodynamic system can be built. Thus the isothermal flow simulation is the base for the thermal flow simulation. The isothermal flow of the system can be approximated by coupling fluid-mass conservation with Darcy's law which relates a gradient in pore pressure P and gravity to the rate of fluid flow. The law of mass conservation can be expressed as

$\left(\begin{array} {cc} Rate \: of \\ accumulation\end{array}\right)$ = $\left(\begin{array} {cc} Rate \\ transported \: in\end{array}\right)$ - $\left(\begin{array} {cc} Rate\\ transported\: out\end{array}\right)$ + $\left(\begin{array} {cc} Rate \: of\\ production\end{array}\right)$

Following this equation of mass conservation, the fluid-flow equations for the gas phase, water phase, and hydrate phase are:

 

$$\frac{\partial }{\partial t} (\phi \rho_g S_g + \phi \rho_w ... ..._{g,h}) = \nabla (v_g \rho_g + v_w \rho_w X_{g,w}) + \dot{m_g},$$ (1)

 

$$\frac{\partial }{\partial t} (\phi \rho_w S_w X_{w,w} + \phi \rho_h S_h X_{w,h}) = \nabla (v_w \rho_w X_{w,w}) + \dot{m_w},$$ (2)

 

$$\frac{\partial }{\partial t} (\phi \rho_h S_h) = - \dot{m_h}.$$ (3)

The subscripts g, w, and h stand for gas, water, and hydrate, respectively. The porosity of the system is denoted by $\phi$, the density of the i-th phase by $\rho_i$, and the saturation of the i-th phase in the pore space by S_i, which varies on a scale from zero to unity. The density of the gas and water phases are pressure and temperature dependent ($\rho_g(P,T)$, $\rho_w(P,T)$), thus incorporating expansion and compression effects under variable pressure-temperature conditions. The density of hydrate is constant. The mass fraction of component c in phase i is denoted by X_c,i (X_g,w is the mass fraction of gas in water, X_w,w the mass fraction of water in water, etc.). These mass fractions can also change as a function of pressure and temperature X_c,i(P,T)). The gas and water phase have a velocity of v_g and v_w. The gas production/depletion rate is included as $\dot{m_g}$, the water production/depletion rate as $\dot{m_w}$, and the hydrate dissociation/growth rate as $\dot{m_h}$. These source/sink terms are related by Yousif et al. (1991):  

$$\dot{m_h} = \dot{m_g} + \dot{m_w},$$ (4)

 

$$\dot{m_g} = \dot{m_h} {M_g \over {N_h M_w + M_g}}$$ (5)

where M_g and M_w are the mole weights of gas and water, and N_h is the hydrate number. The local gas generation/depletion rate caused by hydrate dissociation or growth can be obtained by the Kim-Bishnoi model Kim et al. (1987) and the Englezos model Englezos et al. (1987):

 

$$\dot{m_g} = K_{d/g} A_s (P_e(T) - P)$$ (6)

where K_d/g is a constant characterizing either hydrate dissociation or growth. The specific surface area of the pore volume occupied by gas and water is A_s, and P_e(T) is the equilibrium pressure of hydrate for a certain temperature T. Equation (6) is a kinematic model which is only valid for sufficiently small pressures and temperatures. Given the equilibrium pressure for hydrates at different temperatures, this formula results in the gas production/depletion rates depending on pressure-temperature fields.

The flow-equations (1), (2), and (3) are coupled with Darcy's law:

 

$$v_g = - \kappa_{abs} {\kappa_{rg} \over \mu_g} ( \nabla P - \rho_g g \nabla z),$$ (7)

 

$$v_w = - \kappa_{abs} {\kappa_{rw} \over \mu_w} ( \nabla P - \rho_w g \nabla z)$$ (8)

which relates the velocities of gas and water (v_g and v_w) to the absolute permeability $\kappa_{abs}$, the relative permeability of gas and water $\kappa_{rg}$ and $\kappa_{rw}$, the gradient in pore pressure P and the gravity effect. The viscosities of water and gas are denoted by $\mu_w$ and $\mu_g$. In this first simulation attempt, the capillary pressure has been neglected; I thus assume the same pore pressure for gas and water. The absolute permeability is a function of hydrate saturation (K_abs(S_h)), dependent on the deposition of the hydrate in the pore space Ecker et al. (1995).

For solving the flow system, the following conditions have to be satisfied:

 

S_g + S_w + S_h = 1, (9)

 

X_g,w + X_w,w = 1, (10)

 

X_g,h + X_w,h = 1. (11)

Combining equations (1), (2), and (3) with equations (7) to (11) results in

$\frac{\partial }{\partial t} (\phi \rho_g S_g + \phi \rho_w (1-S_g-S_h) X_{g,w} + \phi \rho_h S_h X_{g,h})$ 

$$= \nabla (- \kappa_{abs} {\kappa_{rg} \over \mu_g} \rho_g (\... ...w} \rho_w X_{g,w} (\nabla P - \rho_w g \nabla z) ) + \dot{m_g},$$ (12)

$\frac{\partial }{\partial t} (\phi \rho_w (1-S_g-S_h) (1-X_{g,w}) + \phi \rho_h S_h (1-X_{g,h}))$ 

$$= \nabla (- \kappa_{abs} {\kappa_{rw} \over \mu_w} \rho_w (1-X_{g,w}) ( \nabla P - \rho_w g \nabla z )) + \dot{m_w},$$ (13)

 

$$\frac{\partial }{\partial t} (\phi \rho_h S_h) = - \dot{m_h}.$$ (14)

Equations (9) through (14) describe the fluid flow of the three phase/ two component gas-water-hydrate system caused by either hydrate dissociation or hydrate growth. The unknowns are pore pressure P and two saturations, in this case gas and hydrate saturation.