Page 122 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

104 Thomas Russell et al.

makes it possible to express the system of Eqs. (3.60 3.62) in dimension-
less form:
@C @C @S s
2 52 ; (3.68)
@T @X @T
C
@S s
5 Λ p ; (3.69)
ffiffiffiffi
@T 2 X
2X @P
5 1: (3.70)
1 1 βφσ ai S s @X
For the undamaged zone, r i , r , r e , the assumption of incompressibil-
ity is no longer used. Although it would be preferred to solve in both
zones for a compressible fluid for accuracy, the impact of both fluid com-
pressibility and formation damage due to fines migration complicates the
derivation of an exact solution. Here, the diffusivity equation determines
the pressure distribution within the reservoir:

@p k 1 @ @p
5 r ; (3.71)
@t φμc p r @r @r
where c p is the compressibility of the fluid.
The initial condition corresponds to the reservoir pressure:

t 5 0:p 5 p res : (3.72)

The outer boundary condition is:
@p
r 5 r e : 5 0: (3.73)
@r
The inner boundary condition is given by Darcy’s law:

@p qμ
r 5 r i : 5 : (3.74)
@r 2πkr i

3.4.2 Analytical solution

All particle detachment occurs on the line T 5 0 and propagates through
the X-T plane along characteristics of slope 1. The equations for these
characteristics are:
X 5 X 0 2 T; (3.75)