[9.2.1.1] This article has introduced theoretical
predictions from eq. (27)
shown in Figure 3 for continuous
mode capillary desaturation that until now seem to have
remained unnoticed within the established traditional
theory of twophase flow.
[9.2.1.2] The predictions illustrated in Figure 3
provide a quantitative basis to discuss deviations
observed in capillary desaturation experiments.
[9.2.1.3] They help to establish limits of validity for the
traditional theory as well as for continuous mode desaturation.
[9.2.1.4] Predictions require precise knowledge of ,
and
emphasizing the importance and need for
reliable special core analysis of high quality.
[9.2.2.1] The efficiency of residual oil recovery during waterflooding depends not only on the balance of forces, but also on other factors, such as the distribution of fluids inside the medium and/or the desaturation protocol. [9.2.2.2] The difference between continuous mode and discontinuous mode desaturation is known in the literature and it may change the critical capillary number (breakpoint) by several decades (see e.g. [40]). [9.2.2.3] The present paper suggests for the first time equally strong differences between the DO/IWI/G-protocoland the DO/WI-protocol. [9.2.2.4] For the latter protocol the breakpoint is sometimes found decades above unity. [9.2.2.5] Further studies of protocol dependence are encouraged to corroborate and clarify such differences and their origin.
[9.2.3.1] If the saturation or desaturation process is experimentally reproducible one expects for the CO/WI- and CO/OI-protocols that [page 10, §0]
![]() |
![]() |
(59a) | |
![]() |
![]() |
(59b) | |
![]() |
![]() |
(59c) |
holds in the limit where and
are both very small.
[10.1.0.1] The saturation
denotes the plateau saturation.
[10.1.0.2] It is seen from Figures 3 as well
as 4 that the plateau saturation
will in general differ from
.
[10.1.0.3] It fulfills either
or
, where
is defined as the zero
![]() |
(60) |
on the capillary pressure curve for secondary imbibition. [10.1.0.4] The actual value is expected to depend on the protocol.
[10.1.1.1] The stationary pore scale pressure fields
(
) are generally assumed to
represent equilibrium pressures
for local thermodynamic equilibrium
although the stationary phase velocities
may be nonzero.
[10.1.1.2] Even at
the pressures cannot
be constant throughout
the pore space
,
because curved interfaces exist in local
thermodynamic equilibrium within
.
[10.1.2.1] The macroscopic capillary pressure
from eq. (24) measured in
experiments for a sample with saturation
at constant flow velocity
is typically measured
using pressure sensors located in the oil and
water reservoirs outside the sample.
[10.1.2.2] These pressure sensors average the local
equilibrium pressure field
over
the surface
of the sensor.
[10.1.2.3] Assuming that the local equilibrium pressures
depend parametrically only
on
, but are independent of
, one has
![]() |
(61) |
where denote the sensor surface located
in the reservoir of phase
.
[10.1.2.4] In [5] a pore scale capillary
pressure
is computed from image analysis
of oil clusters as a surface area weighted
average of mean curvatures over the
fluid-fluid-interface during the imaging
intervals without flow (
).
[10.1.2.5] In other words a formula such as
![]() |
(62) |
with a weighting function was used
to estimate
.
[10.2.0.1] The weighting function is based on estimating the
fluid/fluidinterfacial area
which in turn depends on the method and
parameters of discretizing the interface.
[10.2.0.2] Here
is the estimate
of the local curvature of the interface.
[10.2.0.3] The computed capillary pressures
reported in Figure 2(a) of [5]
range between
and
.
[10.2.0.4] This would suggest a value of
![]() |
(63) |
for the characteristic pressure .
[10.2.0.5] We emphasize that clusters at
much higher local
have been observed in the experiments.
[10.2.0.6] The weighting function
emphasizes the largest
cluster and this cluster had very low mean curvature.
[10.2.0.7] A precise mathematical relation between
and
cannot be given without a rigorous
connection between the microscopic Newton and
Laplace law and the macroscopic generalized
Darcy law (cf. Section VI).
[10.2.0.8] Equations (61) and (62)
above do not establish such a connection
because the domains of integration are disjoint,
i.e.
.
[10.2.0.9] The derivation of macroscopic capillary properties
of porous media expressed through
from
microscopic knowledge of the curvature field
remains a challenge.
[10.2.1.1] Inertial effects and cooperative dynamics of
mesoscaleclusters have been visualized using recent advances in
X-ray microcomputed tomography synchrotron beamlines [46].
[10.2.1.2] The cooperative dynamics is believed to be related to
leading and trailing menisci connected via viscous pressure gradients.
[10.2.1.3] These inertial effects are not visible on the scale of a single pore.
[10.2.1.4] In a porous medium burst-type events are observed
as reported by [47] or [48].
[10.2.1.5] In [46] it is shown that large events
seem to be more frequent than suggested by
simple percolation models
5: Simple percolation models [49], while containing
a diverging length scale at the pecolation transition,
are difficult to apply in the present context, because
of very strong geometric correlations and because
invasion percolation models are limited to the
slow drainage limit.
[10.2.1.6] While a single pore-scale event occurs
over the millisecond time scale [50], i.e. displacement in a
single pore, the occurrence of multiple spatially correlated events
have been observed to decay over the second time scale [46].
[10.2.1.7] These observations might possibly indicate
the emergence of a mesoscopic time or length
scale intermediate between pore size
and system size
.
[10.2.1.8] Recent experimental evidence from [11] suggests that
cluster lengths are flow rate dependent and widely distributed,
ranging from many hundreds of pores down to a single pore.
[10.2.1.9] If such a cluster length or other mesoscopic length and/or time
scale exists, its upper and lower limits are unknown at present.
[10.2.1.10] This challenges also the interpretation of laboratory-based
results.