Page 165 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 143

which can be understood phenomenologically as the number of pore
volumes that need to be injected to establish equilibrium between the
attached concentration and the critical retention function. Using this
parameter, Eq. (3.145) can be expressed in dimensionless form:

ε @S a 5 S cr U; Γ X; Tðð ÞÞ 2 S a X; TÞ: (3.147)
ð
@T
Eq. (3.147), alongside Eqs. (3.94, 3.95, 3.97, and 3.98) outlined in
Section 3.5, form the dimensionless system of equations to be solved.
As before, the equation for salt transport can be solved using the
method of characteristics to obtain:

1; X . T
Γ 5 : (3.148)
0; X , T
Using this result, and integrating Eq. (3.147) as an ordinary differential
equation allows deriving the attached concentration:

S a1 ; X . T
(
S a 5 X2T : (3.149)
ε
S a0 1 S a1 2 S a0 Þe ; X , T
ð
where S a1 and S a0 are the values of the critical retention function when
the dimensionless salinity is 1 and 0 respectively. The primary difference
in the solution procedure for the suspended concentration is the continu-
ity of the solution. In Section 3.5, the suspended concentration experi-
enced a shock along the salinity front (X 5 T). A mass balance condition
was applied across the shock to determine the initial condition used to
solve for C behind this shock. It can be shown that when using nonequi-
librium particle detachment, the suspended concentration is continuous
across the salinity front.
The mass balance condition evaluated across the salinity shock is
(Bedrikovetsky, 1993; Lake, 2010):

½ C 1 S a 1 S s ŠD 5 α C ½Š; (3.150)
where D is the slope of the salinity front and is equal to unity in this case.
Given that changes to C will be finite, Eq. (3.95) indicates that the
strained concentration is continuous:

S s 5 0:
½Š (3.151)