The foam generation in the sand pack is important as it gives the initial bubble size distribution, which then influences the stability behavior. The Blake-Kozeny equation is used to estimate the permeability, k, of the sand pack
)
Figure B1: Micrographs of a C/W foam with 90% v/v CO2 stabilized with 1% v/v 2EH-PO5-EO9 at 24 ˚C and 2000 psia as a function of time. Top image is at 0 min, middle image is at 20 min, and bottom image is at 110 min.
10 µm 10 µm
10 µm
Figure B2: Micrographs of a C/W foam with 90% v/v CO2 stabilized with 1% v/v 2EH-PO5-EO9 at 55 ˚C and 2000 psia with brine (2% w/w NaCl, 0.5% w/w CaCl2, and 0.1% w/w MgCl2) as a function of time. The top image is at 0 min, the middle image is at 20 min, and the bottom image is at 110 min.
Scale bars are included.
25 µm
25 µm 25 µm
Figure B3: Micrograph of a C/W foam with 90% v/v CO2 stabilized with 1% v/v 1-Octanol-PO4.5-EO12 at 24 ˚C and 2000 psia showing the coalescence of bubbles over a 2 sec time interval. Circles connected by arrows show bubble coalescence between frames.
50 µm
where Dsand is the effective spherical diameter of a sand particle and φ is the porosity.232 The porosity of the sand pack is determined by the type of packing of the sand grains.
The sand can pack loosely in a cubic type packing order for which case
pore
where dpore is the spherical diameter of the pore between the sand and φ has a value of 0.476. The sand can pack tightly in a hexagonal close packed order where the following equation can be used with φ value of 0.260.
pore
sand d
D =6.46 [B3]
To account for the non-spherical shape of sand, the dpore value calculated from either equation B2 or B3 has been halved due to the extra packing of the random shapes.
The initial sand pack used as a foam generator was 10.2 cm long with a 3.8 mm inner diameter and was filled with large non-spherical sand particles that were 420-840 µm in diameter. For tight packing for a hexagonal close geometry, k of 37.7-151 darcies (µm2) is calculated and the φ value of 0.26 is used. The dpore estimated for this sand pack is 33-65 µm due to the large size of the sand, and it is denoted the 50 µm pore sand pack.
For the sand pack which was 12.1 cm long with an inner diameter of 7.6 mm filled with 125 µm non-spherical sand, the hexagonal close packing geometry gives a k of 3.3 darcy and a φ value of 0.26. The dpore of this system is about 10 µm, which is a similar average pore size as typical Texas cream carbonate core samples from the Wasan reservoir (about 20 µm); however the k of the core sample is only about 1-3 milidarcies. The large impenetrable portions of the carbonate pore decrease k relative to the sand pack.
Foam generated at low shear rates in porous media is expected to create bubbles that are no smaller than the pores.59 Normally, bubble formation is attributed to lamella divison, snap-off, and leave behind mechanisms that take place at the size of the pores.
However, both the 10 µm and 50 µm pore sand packs form bubble sizes smaller than 1
µm. Estimates of the pore body and throat diameters of the sand packs can be accomplished by following the method of Rossen233 for either a meeting of 8 pore throats or a meeting of 4 pore throats
086 the pore throat, Rsand is the radius of spherical sand in the pack, and i is number of pore throats meeting (either 4 or 8). For non-spherical sand, li,pore, ri,Body, and ri,Throat are halved.
Thus for the 10 µm pore sand pack, the li,pore, ri,Body, and ri,Throat are 31-54, 7.2-13, and 4.6-4.8 µm, respectively. For the 50 µm pore sand pack using a Rsand value of 210 µm, the values are 105-182, 24-44, and 15.6-16.3 µm, respectively. The resulting foam bubble sizes can reach values smaller than even the estimated pore throats, indicating the large shear forces may be present in the sand packs to generate the small bubbles.
To form a bubble, the shear stress must be greater than the Laplace pressure of the bubble. For a bubble with a diameter of 1 µm, assuming a γ of 3 mN/m, the shear stress must exceed 12,000 Pa. The lower γ of the C-W interface relative to the air-water or oil-water interfaces leads to the formation of smaller bubbles with a given shear stress. The magnitude of the shear rate for a non-Newtonian fluid in permeable media flow is estimated using
where Q is the total flowrate, Acs is the cross-sectional area, k is the permeability, and φ is the porosity. Currently, a relationship between the shear rate and shear stress in porous media is lacking, thus the shear rates cannot be used to effectively determine whether the shear stress has been overcome. For the 50 µm pore sand pack, at 6 ml/min the shear rate is 3962 s-1 with k of 37.7 µm2 and φ of 0.26. The 10 µm pore sand pack at 6 ml/min with k of 3.3 µm2 and φ of 0.26 has a shear rate of 3300 s-1. At 1.5 ml/min the shear rate for the 10 µm pore sand pack is only 840 s-1 although typical bubble sizes are smaller than those produced with the 50 µm pore sand pack at 6 ml/min. However, the residence time of the foam in the 50 µm pore sand pack is only 8.6 sec at 6 ml/min, and at 1.5 ml/min for the 10 µm pore sand pack it is 163 sec. The longer time through the sand pack, smaller pores size, and increased number of pores in the 10 µm pore sand pack reduce the cell sizes of the resulting foam.