Page 307 - Enhanced Oil Recovery in Shale and Tight Reservoirs

280 Enhanced Oil Recovery in Shale and Tight Reservoirs

BuckleyeLeverett (1942) solution for the viscous dominated flow. They
derived one general scaling group that can represent many of the previously
defined scaling groups with a different proportionality constant. No assump-
tion is needed to derive such scaling groups other than those needed for
Darcy’s model. No fitting parameter needs to be introduced. Schmid and
Geiger (2013) also showed that spontaneous imbibition can be better charac-
terized by the total volume of the wetting phase imbibed than by the frontal
movement of the wetting phase. Cai and Yu (2012) reviewed many imbibi-
tion equations. Here are listed a few for the convenience of later discussions.
Some of the equations are used to upscale the relationship between the
imbibition recovery and dimensionless time t D in a laboratory-scale to that
in a field scale.

10.2.1 Washburn’s equation
Based on Poiseuille’s law, Washburn (1921) derived an equation to describe
imbibition velocity in a single capillary tube. The velocity equation can be
restated as follows without including the coefficient of slip:

2s cos q

DF þ r 2
dl r
¼ (9.54)
dt 8ml
In the above equation, l is the imbibition distance, t is the imbibition
time, F is the potential, s is the interfacial tension, m is the wetting phase
viscosity, q is the contact angle, and r is the capillary radius. For spontaneous
imbibition, DF is zero. The above equation becomes:

dl s cos qr
¼ (10.1)
dt 4ml
2
The velocity multiplied by pr becomes the imbibition volume in a unit
time. The integration results in an equation to describe imbibed volume
versus time:
2
p s cos qr 5
2
V ¼ t (9.55)
2m
The equation shows that the volume of imbibition of a wetting phase
versus the square root of imbibition time has a linear relationship. However,
as early as 1920, Cude and Hulett (1920) observed that the volume curve
becomes flat at later time. This is because in a porous medium, the pores
have different radii, and the imbibition velocity is proportional to the radius