Page 120 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

102 Thomas Russell et al.

Consider flow with fines mobilization, where the attached fine parti-
cle concentration is given by the maximum retention function σ cr (U).
The analytical form of the maximum retention function for a bundle of
capillaries of equal size has been derived from the torque balance
(Bedrikovetsky et al., 2011a; 2012):
" #
2
U
σ a 5 σ cr UðÞ 5 σ 0 1 2 : (3.57)
U m
The attached concentration in the porous media will begin at some
initial value, σ ai . For sufficiently small fluid velocities, the critical reten-
tion function may lie above this initial value. It follows that for velocities
smaller than some velocity, U i :
σ a 5 σ ai : (3.58)

That is, the attached concentration remains at its initial value.
For axisymmetric flow during production in a petroleum reservoir,
the fluid velocity decreases as the distance from the wellbore increases.
Therefore, at some distance from the wellbore, r i , the fluid velocity will
be equal to U i . No particles will detach at any points further from the
wellbore than this distance. The reservoir can thus be divided into two
zones, the damaged zone, r w , r , r i , and the undamaged zone,
r i , r , r e . Here, r w and r e are the wellbore and drainage radii,
respectively.
In the following formulation, the effect of fines migration on a pro-
ducing well are calculated. The assumption of fluid incompressibility will
only be used for the damaged zone.

3.4.1 Mathematical formulation

First, consider the fluid flow within the damaged zone. Within this
zone, the fluid is assumed to be incompressible. The fluid velocity is
calculated as:
q
U 5 : (3.59)
2πr
The mass balance equation for suspended, strained, and attached parti-
cles in radial coordinates is:
@ @
r ð φc 1 σ a 1 σ s Þ 2 ð rcUÞ 5 0: (3.60)
@t @r