6.2.1
Multiphase flow in porous media
To simulate the multiphase flow in a porous medium, the Navier-Stokes equation along with the conservation of mass and the Allen-Cahn phase field equation have been solved for the intrinsic average of the velocity, u. For convenience, we have adopted the symbol u to replace the usual symbol ⟨u⟩ for the average. We have simulated the flow of two immiscible fluids of different densities (ρg < ρl) and viscosities (µg < µl). The capillary
stress is derived from the excess free-energy of the system accumulated in a thin interface of finite thickness (ϵ) between two fluids. The volume fraction, C, of one of the fluids is used as a phase indicator, which takes a value 1 for one fluid and 0 for the other fluid, where the interface of free-energy is defined by C ∈ [0.5 − ϵ/2, 0.5 + ϵ/2]. In other words, the saturation of one phase is C and the other phase is 1 − C. This representation of the two-phase flow is similar to the volume of fluid method, as well as equivalent to the Buckley-Leverett method. Peaceman & Rachford (1962) presents a detailed derivation of how the individual saturation (si) in the Buckley-Leverett model is equivalent to the phase
indicator function C (see also, Chen et al., 2006c). The density, the viscosity, and the velocity of the phase-field system are given by ρ = ρgC + ρl(1 − C), µ = µgC + µl(1 − C),
The simultaneous flow of a liquid and a gas behaves as an incompressible flow if the corresponding Mach number (the ratio of the velocity to the sound speed) is less than 0.3 (Kundu et al., 2012; Antanovskii, 1995). Thus, the conservation of mass becomes
∇ · u = 0.
Since the length scale characterizing the pore structures of the porous medium is much less than that of the REV, as well as the computational mesh, the porous medium is represented by a packed spheres. In this approach, the viscous drag exerted by the spheres is equivalent to the volumetric flow rate that is linearly related to the pressure gradient in the Darcy’s equation (Bear, 1972), and the form drag exerted by the spheres is equivalent to the resistive force in the Forchheimer equation (deLemos, 2006). For each phase, the porous medium is modelled in terms of the total resistive (drag) force
fi = − [ ν K+ CF |u| √ K ] u,
where K[m2] is the effective permeability of the model porous medium, ν is the kinematic viscosity and CF is a constant. For spheres of diameter d (e.g. characteristic pore-scale),
both K and CF can be estimated using Ergun’s methodology (Ergun, 1952) such that CF =
F/√K, K = d 2ϕ3 (1 − ϕ)2, F = 1.75d 150(1 − ϕ),
and ϕ is the porosity of a porous medium. A technical details of the above model for single- phase flow in porous media is documented by deLemos (2006), Breugem & Rees (2006) and Bear (1972). Following Antanovskii (1995) and Breugem & Rees (2006), we have derived the following ‘intrinsic average’ of the momentum equation for a two-phase flow in porous media: ρDu Dt = −∇P + ∇ · (µ∇u + µ∇u T) −ϕµu K − CFϕ|u|u √ K + (ρl− ρg)g − ∇ · τσ, (6.1)
where
Du Dt =
∂u
∂t + u · ∇u,
∇uT denotes the transpose of the velocity gradient tensor ∇u, and τ
σ = σκϵ2∇C ⊗ ∇C
denotes the stress tensor contributed by the surface tension.
6.2.2
The phase-field method and the surface tension
6.2.2.1 Dynamics of the two-phase interface
The phase-field method is a relatively new approach for two-phase flow simulation (Vas- concelos et al., 2014; Kim, 2012; Provatas & Elder, 2011; Feng et al., 2007; Jacqmin, 1999; Antanovskii, 1995). It employs the Navier-Stokes equations to accurately simulate the motion of a contact line (Cai et al., 2015) or to directly simulate pore-scale flow of two fluids (Alpak et al., 2016). There are two versions of the phase-field method. The Cahn- Hilliard method uses a fourth order diffusion (e.g. Jacqmin, 1999) and the Allen-Cahn method uses a second order diffusion (e.g. Vasconcelos et al., 2014). Both approaches pre- serve the immiscibility of fluids by conserving mass of each phase. This article has adopted the Allen-Cahn method because its numerical treatment is more efficient with respect to the Cahn-Hilliard method.
In phase-field methods, the total free-energy is defined by the functional
W = ∫ Ω ( ϵ2 2|∇C| 2+ f (C) ) dΩ, (6.2)
where ϵ is the thickness of the interface containing excess free-energy of the two phase system, and the double well potential takes the form f (C) = C2(1 − C)2. The dynamics
of the interface is then governed by the evolution of C(x, t) according to ∂C ∂t + u · ∇C = T µ c+ T |Ω| ∫ f′(C)dΩ, (6.3)
where 1/T is called the elastic relaxation time-scale and µcis called the chemical potential, which is defined by
µc= −δW δC = ϵ
2∇2C − f′(C)
is called the chemical potential (Yang et al., 2006). The last term in (6.3) ensures the conservation of C (Vasconcelos et al., 2014). Clearly, the chemical potential, µc, vanishes
away from the interface. In other words, the phase-field equation (6.2) approaches to the VOF equation away from the interface (Alpak et al., 2016; Hirt & Nichols, 1981).
6.2.2.2 Surface tension
The surface tension across a fluid-fluid interface can be obtained from the Young-Laplace equation
Pg − Pl = σκ,
where σ [N/m] is the surface tension and κ [1/m] is the local curvature of the inter- face. (Chen et al., 2006a). Thus, the stress acting on an arbitrary fluid surface must account for an additional component σκ in a two-phase system in such a way that the surface ten- sion does not alter the stress tensor in the region that is away from the fluid-fluid interface. An accurate representation of stress in a two-phase system is a potential advantage of the phase-field theory. Antanovskii (1995) and Jacqmin (1999) provide a detailed derivation of stress that is modified by the surface tension. For the present Allen-Cahn method (cf. Vasconcelos et al., 2014), the assumptions remain the same as the assumptions considered by Jacqmin (1999) for the Cahn-Hilliard method. In the Navier-Stokes equations, the total stress for a two-phase flow is thus given by (e.g. Eq 6.1)
∇ · τ = ∇ ·[−P I + µ (∇u + ∇uT)] + σκµc∇C,
where the last term accounts for the divergence of stress due to the surface tension, which has been derived using the variational derivative of W (cf. eq. 6.2) and expanding ∇ ·
(∇C ⊗ ∇C). In other words, the last term in eq (6.1) is given by ∇ · τσ = −σκµc∇C.
We have employed the above theory to simulate a two-phase flow of a gas and a liquid by solving Eqs (6.1-6.2) along with the conservation of mass, ∇ · u = 0.