Page 108 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

90 Thomas Russell et al.

The final form of the critical retention function versus velocity and
salinity will not only roughly follow those shown in Fig. 3.8, but will also
be highly dependent on the form of the particle size distribution.

3.2.3 Single-phase equations for fines transport
Mathematical modeling of fines migration typically revolves around a
continuity equation for the suspended particle concentration. The form
of this equation is as follows:

Rate of accumulation 5 Divergence of advective flux
Rate of detachment Rate of straining:

Suspended particle transport is given by the sum of the advective and
diffusive flux. For the purposes of modeling fines migration in petroleum
reservoirs, diffusive flux is typically negligible and thus is removed for
convenience. Thus, the continuity equation can be expressed as:
@φc @c @σ a @σ s
52 U 2 2 : (3.22)
@t @x @t @t
This is more commonly given as:
@ @c
½ φc 1 σ a 1 σ s Š 1 U 5 0 ; (3.23)
@t @x
where c, σ a ,and σ s are the concentrations of suspended, attached, and
strained particles, respectively, φ is the porosity of the porous media, U is
the fluid flow velocity, x is the distance, and t is the time. The porosity
term arises because the suspended concentration is defined as the volume
of particles in suspension per pore volume, while the attached and strained
concentrations are defined similarly but for the bulk rock volume.
For Eq. (3.23) to be valid, it is necessary here to assume that during
the transport, detachment, and capture, the particle and fluid volumes are
additive (Amagat’s law), and c, σ a , and σ s are the volumetric concentra-
tions. Eq. (3.23) is also valid where the mass concentrations of all particle
species are negligibly small if compared with the mass of water.
The straining rate is assumed to be proportional to particle advection
flux, cU (Bedrikovetsky, 2008; Herzig et al., 1970):

@σ s
ðÞcU;
5 λσ s (3.24)
@t
where λ is the filtration coefficient for straining.