Pore Network Investigation of Gas Trapping and Mobility During Foam Propagation Using Invasion Percolation with Memory

Abstract

Foam reduces the gas mobility in porous media both by increasing the effective viscosity of the gas phase and by trapping a large portion of the gas in place. This reduction is directly related to the number density of lamellae in the gas phase. Therefore, understanding the pore-level events associated with lamella generation and destruction processes and investigating the trapped foam behavior are of great importance in modeling foam mobility. In this paper, a pore network model based on the statistical physics method of invasion percolation with memory (IPM) is developed to simulate foam propagation as a drainage process of gas invasion into a porous media initially saturated with a surfactant solution. During this process, lamella generation, destruction, and mobilization are involved. This study sets out to explore the roles of pore level events that lead to foam destruction. To do so, static lamella destruction by capillary suction at the plateau borders is modeled using the Reynolds equation for film thinning and lamella rupture is assumed to occur when the film thickness falls below a certain critical thickness (h_fc) at which the maximum disjoining pressure (Π_max) is attained. This mechanism is incorporated in the pore network model to which we add a notional time dependency of the invasion percolation with memory mechanism. Flowing lamellae are assumed to rupture at a fixed limiting capillary pressure (P^*_cap ) lower than Π_max. Results show that a critical regeneration probability (f^*_reg ) is required for the generation of strong foam in the network. The mobilization pressure gradient depends on both the number of lamellae in the flow path and the sizes of the throats that make up of this path. At the same f_reg, the mobilization pressure gradient markedly decreases after incorporating lamella destruction mechanism. The structure of the displacement pattern of the invading phase at breakthrough changes under the competition between capillary and yield stress-like forces. During transient foam displacement, gas saturation increases, and foam texture becomes finer with increasing f_reg. The flowing foam fraction increases much more slowly with pressure gradient after accounting for the viscous friction associated with the flow in the already open paths. Comparison with experiments shows that current pore network model can capture the main features of the transient foam flow in porous media.

Appendix

In order to determine the capillary number at the end of each invasion stage, we should first determine the corresponding global velocity field.

If a throat has no lamella, the gas phase is still continuous and behaves like a Newtonian fluid. As an approximation, we treat pore throats as if they were cylindrical and obtain the flow rate and the pressure drop relation for Newtonian fluids such as the continuous gas phase and the liquid phase using a simple Poiseuille flow equation (Chen 2005a, b):

$$q_{ij} = \Delta p_{ij} r_{ij}^{4}.$$

(27)

If a throat on the minimum threshold path (MTP) contains a lamella, the gas phase behaves as though it had yield stress. The flow rate and pressure drop relationship is illustrated by Chen et al. (2005a, b):

$$q_{ij} = r_{ij}^{4} \left( {1 - \frac{1}{{\Delta p_{ij} r_{ij} }}} \right)\Delta p_{ij} , {\text{when }}\left| {\Delta p_{ij} } \right| > \frac{1}{{r_{ij} }}.$$

(28)

Equations (27) and (28) are non-dimensionalized by using the average throat radius, R_m, as the characteristic length, 4γ/R_m as the characteristic pressure, and πγR²_m /2 μ_g (or πγR²_m /2 μ_l) as characteristic flow rate. μ_g is the viscosity of gas in the absence of foam (0.02 cp at 298 K), and μ_l is the liquid viscosity (0.9 cp at 298 K) (Chen et al. 2006).

We assume that the fluid is incompressible; a material balance equation is written for each pore under considerations, including pores occupied by liquid, pores along the MTP, and pores along the already open paths after breakthrough:

$$\mathop \sum \limits_{j}^{Z} q_{ij} = 0$$

(29)

where Z is the coordination number and equal to 4.

To be more specific, as shown in Fig. 21, if pores i and j are both occupied by liquid (D₃ and D₅ in Fig. 21), Eq. (27) is applied to calculate the liquid flow rate through the pore throat connecting them, in which πγR²_m /2 μ_l is used as the characteristic flow rate. If a pore is currently at the front and on the MTP and the other is occupied by liquid (M₄ and D₃ in Fig. 21), the flow rate through the throat connecting them is given by:

Fig. 21

Simplified schematic of a gas invasion stage. Blue and yellow represent the gas and the liquid phase, respectively. Pores with red circle denote the pores on the MTP at the current stage

$$q_{ij} = \left( {p_{i} - p_{\text{cap}} - p_{j} } \right)r_{ij}^{4}.$$

(30)

If pores i and j are both occupied by gas and on the MTP, and no lamella is present in the throat connecting them (M₂ and M₃ in Fig. 21), Eq. (27) is applied, in which πγR²_m /2 μ_g is used as the characteristic flow rate. Otherwise, if a lamella is present (M₁ and M₂ in Fig. 21), Eq. (28) is applied. No-flow boundary condition is applied to the pores adjacent to the front (D₁ and D₂ in Fig. 21), except those adjacent to the MTP (D₃ and D₄ in Fig. 21). Constant-pressure boundary condition is applied to pores along the inlet and outlet of the lattice.

After inserting the appropriate local expression in Eq. (29), the resulting equations are solved by a combination of conjugate gradient and over-relaxation methods (Batrouni and Hansen 1988; Hadjidimos 2000).