The expression model
Due to the distinction between inter-aggregate and intra-aggregate
liquid, we need two continuity equations with a source and sink term,
respectively, at the RHS. Thanks to the use of the material coordinateω , we arrive at simple continuity equations for the
inter-aggregate oil
and for the intra-aggregate oil
respectively, in which q in the source/sink terms at the RHSs
denotes the release, per aggregate volume (in s-1), of
inter-aggregate oil from the aggregates. We have dropped the convective
term in Eq. (1.4), since as long as the liquid stays within the
aggregate pores, its velocity (relative to the solids) is zero. In Eq.
(1.4), the total solidosity occurs within the time derivative. Our
biporous model essentially differs from the simple single continuity
equation ∂e/∂t = ∂u/∂ω used by Sørensen et al. [24]
and Kamst et al. [21].
The (local) flux u depends on the (local) pressure gradient in
the liquid phase and is assumed to obey Darcy’s law with permeabilityk . The convective term of Eq. (1.4) is then rewritten:
The liquid pressure balances the stress in the deforming filter cake
(see e.g. , Olivier et al. [9]) :
while an elastic modulus E connects the solids pressureps with the logarithmic strain:
with δ standing for the thickness of the filter cake and the
subscript 0 denoting initial values, before cake deformation sets in.
Applying the chain rule twice, using Eqs. (1.7) and (1.8), and
eliminating the total solidosity s results in
We should realize that in a non-linearly elastic medium the elastic
modulus depends on the filter cake strain itself, i.e. E =E (e 1, e 2). These
manipulations turn the (seemingly) convective term of Eq. (1.4) into a
diffusive term. Such a diffusive term is not uncommon: see e.g. ,
Tosun [26], Sørensen et al. [24], Kamst et al. [21], and
Olivier et al. [9]. As a matter of fact, the basic idea can already
be found in the classical Terzaghi paper dated as early as 1923 [6].
Substituting Eq. (1.9) into Eq. (1.4) and re-writing the solidositiess and s 2 in terms ofe 1 and e 2 results in
while Eq. (1.5) can be rewritten as
The next step is to find an expression for the release rate q .
Different from Mrema and McNulty [19], we assume the flux out of the
aggregates is Darcian, with a permeability k 2 =k 2(e 2) associated with the
aggregates, through the specific area a =
6/da for the spherulitic aggregates of constant
average size da . The pressure gradient can be
transformed as above, resulting in
The above Eqs. (1.10) and (1.12) contain the cake propertiesk 1, k 2 and E which
all are dependent on the pertinent the pertinent void ratios. We need
empirical correlations for these parameters. As, according to Tien and
Ramarao [23], the Kozeny-Carman relation is not valid under
consolidating conditions, we use the Meyer and Smith [27]
correlation
For k 1, we use void ratioe 1 and aggregate size da ,
while k 2 needs e 2 and the
typical diameter dc of the individual crystals
that build the agglomerate. Fitting an exponential function through data
for strainmeasured at varying constant load ppresults in an expression of the type
Using Eq. (1.8) then results in the expression
The eventual set of the two partial differential equations fore 1 and e 2 then is
in which
Ce is a type of diffusion coefficient, in the
consolidation literature denoted as a modified consolidation coefficient
[9, 24]. While this coefficient in a real-life expression process is
varying with position and in time, in many papers (e.g. ,
[14], [28]) it is treated as a constant: this simplifies solving
the consolidation equation which is a second-order partial differential
equation. Kamst et al. [21], however, appreciate the consolidation
coefficient (also) depends on local cake porosity and compressibility.
The review paper by Olivier et al. [9] cites a number of authors
(among which [5]) who all use similar relationships for diffusivity
or consolidation coefficient. Our expression forCe in Eq. (1.18) is essentially different from
earlier proposals due to the biporous character of our fat crystal
slurry as a result of which it includes both the intra-aggregate and the
inter-aggregate solidosities. In addition, our consolidation equation,
Eq. (1.16), contains a source term which to the best of our knowledge is
a novelty. Finally, our model looks much simpler than Lanoisellé’s.
In more general terms, our expression model is a rheological model
composed of two dashpots in series parallel to a spring. The double
porous nature of the fat crystal aggregate filter cake is represented as
a series of two dashpots described with the Meyer & Smith correlation
for the permeability (rather than the Kozeny-Carman relation). The
spring is due to the elastic modulus that can be determined
experimentally with a constant load test.