Page 125 - Formation Damage during Improved Oil Recovery Fundamentals and Applications
P. 125
Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 107
zone 3. Substituting the expression for the maximum retention function
(Eq. (3.57)) enables the evaluation of the integral:
1 h p ffiffiffiffi p ffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffi i
S s X;Tð Þ5 p ffiffiffiffi Λ X 112 Λ X m 11 e Λð X2 X mÞ
Λ X
h p p ffiffiffi p p p ffiffiffi p ffiffiffiffiffiffiffiffi i
σ ai 2σ 0 ffiffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 p ffiffiffiffi Λ X m 11 e Λð X2 X mÞ 2 Λ X 1T 11 e Λð X2 X1TÞ
φσ ai Λ X
ð T 2 p
ffiffiffiffiffiffiffiffi
Λσ 0 ΛX qT ðÞ 2Λ X1T
1 p ffiffiffiffie e dT:
2 2
2
8π r U φσ ai X X m 2X X 1T
i m
(3.81)
The solution is presented here for an arbitrary injection rate, q(T).
The solution for zone 3 is similar, with the exception that the initial
condition corresponds to the absence of strained particles along the line
T 5 0.
p ffiffiffih p p p ffiffiffi p ffiffiffiffiffiffiffiffi i
σ ai 2 σ 0 Λ X ffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi Λð X2 X1TÞ
S s X; Tð Þ 5 p ffiffiffiffi e Λ X 1 1 2 Λ X 1 T 1 1 e
φσ ai Λ X
ð T 2
p
ffiffiffiffiffiffiffiffi
Λσ 0 ΛX qTðÞ 2Λ X1T
1 p ffiffiffiffi e e dT:
2 2
2
8π r U φσ ai X 0 X 1 T
i
m
(3.82)
All characteristics comprising zone 4 (X w , X , X m , X 1 T . X i ) and
zone 5 (X m , X , X i , X 1 T . X i ) begin on the X-axis at X values greater
than X i . As such, the suspended concentration is zero in these zones.
The strained concentration in zone 4 consists of particles strained in
zones 1 and 2. The strained particle concentration in zone 5 is equal to
the total concentration of particles that strained in zone 3. As such, the
strained concentration for these two zones will be given by:
ð
S s X; Tð Þ 5 S s X; X i 2 XÞ; (3.83)
and will be independent of time.
Zone 6 lies entirely outside of the damaged zone. As such, there are
no suspended or strained particles in this zone.
CX; TÞ 5 S s X; TÞ 5 0: (3.84)
ð
ð
Table 3.4 presents the summary for all the solutions derived in this
section.
