Effective viscosity of foam in periodically constricted tubes

The mobility of gas in foam in porous media depends both on an effective yield stress and on the drag of moving bubbles on pore walls. Previous modeling has split the effective gas viscosity into those two parts and solved for them separately. For the first time we present a dynamic model for the movement of bubbles through constricted tubes in 2D, accounting for the drag on lamellae along pore walls and the capillary forces that govern bubble shape in periodically constricted tubes. In the limit of slow flow rate, lamellae jump to asymmetric shapes even in radially symmetric pores, in agreement with Rossen [J. Colloid Interf. Sci. 136 (1990) 1; J. Colloid Interf. Sci. 136 (1990) 17; J. Colloid Interf. Sci. 136 (1990) 38; J. Colloid Interf. Sci. 139 (1990) 457]. The drag on the lamella increases the pressure gradient above the quasi-static limit by a factor scaling roughly as the 2/3 power of velocity, in agreement with Hirasaki and Lawson [SPE J. 25 (1985) 176]. At sufficiently high velocities, however, a symmetric jump replaces the asymmetric jump. In such cases unstable perturbations away from the symmetric shape have insufficient time to grow before the lamella settles into a symmetric shape on the other side of pore body. A lamella moving at a fast rate may require a lower pressure gradient than one moving more slowly, due to the disappearance of the asymmetric jump. Observations of lamella movement in glass pores support these predictions.