Page 134 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

114 Thomas Russell et al.

Eq. (3.98) after solving the system of Eqs. (3.94 3.96). The exact solu-
tion for 1D flow with salinity alteration and fines migration is derived
below. The solution is based on the mass-balance condition on a shock
front, and on the method of characteristics (Polyanin and Zaitsev, 2011;
Polyanin and Manzhirov, 2007).
Eqs. (3.96 and 3.97) for attached concentration and mass balance of
salt transport both separate from the rest of system (Eqs. (3.94 3.98)) and
can be solved immediately using the Method of Characteristics subject to
the initial (Eq. (3.99)) and boundary (Eq. (3.100)) conditions. It follows
that the salinity is equal to the formation water salinity ahead of the salin-
ity front and equal to injected water salinity behind the front. This is
shown in the dimensionless form in Eq. (3.101). The solution for the
attached concentration corresponds to the critical retention function at
the two salinities behind and ahead of the shock (Eq. (3.102)):

1; X . T
Γ 5 ; (3.101)
0; X , T
(
S cr γ ; X . T
I
S a 5 : (3.102)
S cr γ ; X , T
J
In all zones, the salinity values are constant, so the time derivative of
attached concentration is equal to zero. Substituting the straining rate
(Eq. (3.95)) into Eq. (3.94) yields:
@C @C
1 α 52 αΛC: (3.103)
@T @X

The above Partial Differential Equation (PDE) is solved using the
method of characteristics in three different zones: ahead of the salinity
front (X . T), between the salinity front and the suspended particle front
(αT , X , T), and behind the suspended particle front (X , αT).
Fig. 3.16A shows the three different zones indicated by the numbers 0, 1
and 2, respectively.
For X . T, zone 0, along the parametric curves given by:
dX
5 α; (3.104)
dT
Eq. (3.103) reduces to the ordinary differential equation:

dC
52 αΛC: (3.105)
dT