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Gas and Liquid Injection Rates 185

and

n = 0:095932 + 2:3654Γ − 10:467Γ + 12:955Γ 3
2
(9.19)
5
4
+ 14:467Γ − 39:673Γ + 20:625Γ 6
where the in situ foam quality index Γ is defined as

2116
Q go
Γ = P (9.20)
2116 Q go + 1 Q m + 5:615
P 7:48 60 Q f

or
Γ = GLR (9.21)
1 + GLR
It is understood that the K-Γ relation and the n-Γ relation given by
Eqs. (9.18) and (9.19), respectively, can be different for different types of
foams (foams formed by different gases, liquids, and chemicals).
In the range of foam quality for the Bingham plastic model, the effec-
tive foam viscosity can be estimated by

μ = μ + g c τ o D H (9.22)
e p
6v f
where μ p = plastic viscosity, and τ o = yield point.

Illustrative Example 9.2
A well was cased to 5,280 ft with an 8-⅝-in, 28-lb/ft (8.017-in ID) casing.
Starting from the kickoff point at 5,315 ft, the hole was drilled with mud
using a 7⅞-in bit to build an inclination angle at a constant build rate of
4°/100 ft until the maximum inclination angle of 90 degrees was reached at a
depth of 7,558 ft. Then the drilling fluid was shifted from mud to stable foam
to drill to the TD of 8,048 ft while the inclination angle was maintained at
90 degrees. Additional data are given as follows. Calculate and plot the pro-
files of pressure, velocity, foam quality index, mixture density, and ECD along
the flow path when the bit was at the total depth.
(Continued )