Impact of Two-Phase Flow Pattern on Solvent Vapour Extraction

Abstract

Solvent vapour extraction (Vapex) is a promising technology for in-situ heavy oil recovery from oil sands deposits. The prediction of oil recovery rates requires fundamental understanding of the pore-scale mechanisms and their impact on mass transfer and oil production. To bridge the gap between pore-scale mechanisms and macro-scale recovery, a dynamic pore-network model for two-phase flow with mass transfer is developed. The impact of pressure gradient on two-phase flow pattern, mass transfer and oil production are investigated. It is found that at high capillary numbers, in viscous dominated flow, dissolved oil is moved in intermittent liquid clusters to the outlet of the network. This mechanism of interface renewal maintains a steep solvent mole fraction gradient at the interface and enhances mass transfer, resulting in high oil production. In capillary-dominated flow, capillary fingering with low mass transfer and oil production are observed.

Appendices

Appendix A: Local Rules

The analytical expressions used for the calculation of capillary pressures in pores, hydraulic conductivities for liquid and gas phases and conditions for throat invasion and snap-off are presented in this appendix.

1.1 A.1 Pore Capillary Pressure

Pore bodies are modelled as cubes with the wetting phase residing in the corners and along the edges. The Capillary pressure in a pore body, ${p}_{i}^{c}$, depends on the inscribed radius, R_i, and liquid saturation, ${S}_{i}^{l}$, following the equation (Joekar-Niasar et al. 2010):

$$p_{i}^{c} \left( {s_{i}^{l} } \right) = \frac{2 \sigma }{{R_{i} \left( {1 - \exp \left( { - 6.83 s_{i}^{l} } \right)} \right)}},$$

(A.1)

where $\sigma$ is the interfacial tension between wetting and non-wetting fluids.

A minimum liquid-phase saturation is associated with each pore body, $s_{{i, {\text{min}}}}^{l}$. It depends on the global pressure difference imposed between the inlet and outlet of the model, $p_{{{\text{global}}}}^{c}$. It is given by the following equation:

$$s_{{i,{\text{min}}}}^{l} = - \frac{1}{6.83}\;\ln \left( {1 - \frac{1}{{R_{i} }}\frac{2 \sigma }{{p_{{{\text{global}}}}^{c} }}} \right).$$

(A.2)

1.2 A.2 Throat Invasion

The vapour phase invades a pore throat when the capillary pressure in the neighbouring pore body exceeds the entry capillary pressure of the pore throat. It is given by the following equation (Joekar-Niasar et al. 2010):

$$p_{e, ij}^{c} = \frac{\sigma }{{r_{ij} }} \left( {\cos \theta + \sqrt {\frac{\pi }{4} - \theta + \sin \theta \cos \theta } } \right),$$

(A.3)

where $r_{ij}$ is the radius of the inscribed circle of the pore throat cross section, and $\theta$ is the contact angle.

1.3 A.3 Snap-Off

When the capillary pressure in a pore throat drops below a critical value, liquid in the corners snaps-off and fills the throat. For a throat with square cross section, snap-off occurs when (Vidales et al. 1998):

$$p_{ij}^{c} \le \frac{\sigma }{{r_{ij} }} \left( {\cos \theta - \sin \theta } \right).$$

(A.4)

1.4 A.4 Throat Conductance

When the throat is occupied by a single phase $\alpha$, phase conductance is given by the following equation (Patzek 2000):

$$K_{ij}^{\alpha } = \frac{{0.5623 G_{ij} \left( {A_{ij} } \right)^{2} }}{{\mu^{\alpha } L_{ij} }},$$

(A.5)

$G_{ij} = \frac{{A_{ij} }}{{\left( {C_{ij} } \right)^{2} }}$, is the shape factor of the pore throat cross section. $A_{ij}$, $C_{ij}$ and $L_{ij}$ are the cross-sectional area, the perimeter and the length of the throat connecting pores i and j. The length of the throat connecting pores i and j is assumed to be equal the distance between the centre of the pores.

When a pore throat is invaded and two-phases coexist along the throat as shown in the figure above, the conductivity of the phase is given by the following equation:

$$K_{ij}^{g} = \frac{{0.5623 G_{ij}^{g} \left( {A_{ij}^{g} } \right)^{2} }}{{\mu^{g} L_{ij} }},$$

(A.6)

$G_{ij}^{g} = \frac{{A_{ij}^{g} }}{{\left( {C_{ij}^{g} } \right)^{2} }}$, is the shape factor of the non-wetting phase region in the pore throat cross section; $A_{ij}^{g}$, is the cross-sectional area of the gas phase region, given by:

$$A_{ij}^{g} = 4 R_{ij}^{2} - 4 r_{ij}^{2} \left[ {\frac{{\sin \left( {\frac{\pi }{4} - \theta } \right)}}{{\sin \left( {\frac{\pi }{4}} \right)}}\cos \left( \theta \right) - \left( {\frac{\pi }{4} - \theta } \right)} \right],$$

(A.7)

where

$$r_{ij} = \frac{{p_{ij}^{c} }}{\sigma }$$

(A.8)

The perimeter of the non-wetting phase region: $C_{ij}^{g}$ is given by the following equation:

$$C_{ij}^{g} = 8 \left[ {R_{ij} - r_{ij} \left( { - \frac{{\sin \left( {\frac{\pi }{4} - \theta } \right)}}{{\sin \left( {\frac{\pi }{4}} \right)}} + \left( {\frac{\pi }{4} - \theta } \right)} \right)} \right]$$

(A.9)

The conductance of the wetting liquid phase in the corners of the throat is computed using a semi-empirical model with perfect slip boundary condition between liquid and gas phases (Sidian 2020):

$$K_{ij}^{l} = 4\frac{{2 \tilde{g}_{ij} l_{ij}^{4} }}{{\mu_{ij}^{l} L_{ij} }}$$

(A.10)

$l_{ij}$, is the meniscus-apex distance along the wall at the corner. It is given by the following equation:

$$l_{ij} = r_{ij} \frac{{\cos \left( {\theta + \beta } \right)}}{\sin \left( \beta \right)}$$

(A.11)

$\beta$, is half corner angle. For a throat with square cross-sectional area, $\beta = \frac{\pi }{4}$. The dimensionless conductance of the wetting phase is given by:

$$\tilde{g}_{ij} = {\text{exp}}\left\{ {\frac{{\left[ { - 18.2066 \left( {\tilde{G}_{ij}^{l} } \right)^{2} + 5.88287 \tilde{G}_{ij}^{l} - 0.351809 + 0.02 {\text{sin}}\left( {\beta - \frac{\pi }{6}} \right)} \right]}}{{\left( {\frac{1}{4\pi } - \tilde{G}_{ij}^{l} } \right)}} + 2\ln \tilde{A}_{ij}^{l} } \right\}$$

(A.12)

$\tilde{A}_{ij}^{l}$ and $\tilde{G}_{ij}^{l}$ are the area and shape factor of the wetting phase region at a corner with a unit meniscus-apex distance, respectively. $\tilde{A}_{ij}^{l}$ and $\tilde{G}_{ij}^{l}$ are given by:

$$\tilde{A}_{ij}^{l} = \left[ {\frac{\sin \left( \beta \right)}{{\cos \left( {\theta + \beta } \right)}}} \right]^{2} \left[ {\frac{{\sin \left( \beta \right)\cos \left( {\theta + \beta } \right)}}{\sin \left( \beta \right)} + \theta + \beta - \frac{\pi }{2}} \right]$$

(A.13)

$$\tilde{G}_{ij}^{l} = \frac{{{ }\tilde{A}_{ij}^{l} { }}}{{4\left[ {1 - \left( {\theta + \beta - \frac{\pi }{2}} \right)\sin \left( \beta \right)/{\text{cos}}\left( {\theta + \beta } \right)} \right]^{2} }}.$$

(A.14)

Appendix B: Algorithm

The algorithm used for solving the molar balance equation and using the fully implicit method is presented in this appendix.

2.1 Algorithm Description

• 1. Input data and generation of the pore network structure

• 2. Set initial condition, t = 0, for: $S^{l}$,$x^{{{\text{solv}}}}$,$P^{l}$ in the pores and initial time step, $\Delta t$.

• 3. Define the solution vector, the primary variable vector $X = \left[ {X_{i}^{p} } \right]_{{i = 1,N_{{{\text{pore}}}} }}$, where: For a single liquid-phase pore i, the primary variables are solvent mole fraction and pressure in the liquid phase $X_{i}^{p} = \left[ {x_{i}^{{{\text{solv}}}} P_{i}^{l} } \right]$. For a two-phase pore i, the primary variables are liquid saturation and pressure: $X_{i}^{p} = \left[ {S_{i}^{l} P_{i}^{l} } \right]$.

• 4. While t < t_end do

  • a. Initialization of nonlinear iteration loop count: iter = 0

  • b. While $X^{t + \Delta t,iter + 1} - X^{t + \Delta t,iter}_{2} > tol$ and iter < max_iterations do

  • c. Update throat occupancy based on pore capillary pressures and local rules for throat invasion and imbibition

  • d. Calculate liquid and gas phase conductivities in the throats and their saturation and derivatives with respect to solvent mole fraction and liquid saturation

  • e. Assemble the Jacobian matrix Jac and right-hand side vector F

  • f. Solve the linear system $Jac \times \Delta X = - F$

  • g. Update the primary variable vector: $X^{t + \Delta t,iter + 1} = X^{t + \Delta t,iter} + \Delta X$ and the corresponding liquid-phase saturation, solvent mole fraction and pressure at each pore.

  • h. For a pore under imbibition and liquid saturation sufficiently close to 1.0, make the pore single phase and the throats connected to it all imbibed with the liquid phase

  • i. For a pore under drainage and liquid saturation sufficiently close to $S_{i min}^{l}$. Fix liquid pore saturation to $S_{i min}^{l}$ in the time step.

  • j. For pores near phase transition, run flash calculation and determine if the pores are in single or two-phase state. Calculate the saturation of the liquid phase for a two-phase pore.

  • k. Check for convergence of the local newton iteration while checking if the maximum pore saturation change during the iteration exceeds a limiting value,0.2 in this work. If that is the case, reduce the time step by half and restart the iterations at (b).

  • l. When the local newton iterations have converged. Calculate the maximum relative change of pore saturation during the time step. If it exceeds a limiting value, 0.2, reduce the time step by half and restart the iterations at (b).

  • m. Estimate time step $\Delta t$ for the next time step $t = t + \Delta t$ using the calculated rate of local liquid drainage or imbibition in the pores. Go back to (a) until t = t_end.

Appendix C: Time Step

The time step is estimated using the calculated rate of local liquid drainage or imbibition in the pores and change in solvent mole fraction. Local time step $\Delta t_{i}$ in pore i is calculated as follows:

• For two-phase pore i:

  $$\Delta t_{i} = \left\{ {\begin{array}{*{20}l} {\left| {\frac{{\left( {1 - S_{i}^{l} } \right)}}{{q_{i}^{n} }}} \right|, } \hfill & \quad {q_{i}^{n} < 0} \hfill \\ {\left| {\frac{{\left( {S_{i}^{l} - S_{i \min }^{l} } \right)}}{{q_{i}^{n} }}} \right|,} \hfill &\quad {q_{i}^{n} > 0} \hfill \\ \end{array} } \right.$$

  (C.1)

The time step is obtained as follows:

$$\Delta t = \min \left\{ {\left( {\Delta t_{i} } \right)_{{i = 1,N_{{{\text{pore}}}} }} ,2\Delta t_{0} } \right\},$$

(C.2)

where: $\Delta t_{0}$ is the previous time step.